 understand the reality described by QM, we will be following the math of partitional concepts. The development of the mathematical concepts from the logical level up to QM is based on the natural bridge between set concepts and vector-space concepts using the notion of a basis set. We will transport partitional concepts across that bridge in both directions. We will see that the mathematics and logic of set partitions can be lifted or transported to complex (inner product) vector spaces where it yields essentially the mathematical machinery of QM (of course, not the specifically physical postulates such as the Hamiltonian or the DeBroglie relationships). This shows that the mathematical framework of QM is a type of logical system developed using complex vector spaces. The vector space concepts of full QM can also be transported back to set-like vector spaces over Z2 to yield a pedagogical model of "quantum mechanics over sets" or QM/sets [15]. The probability calculus of QM/sets is a noncommutative version of the usual Laplace-Boole logical finite probability theory. The traffi c in both directions supports the idea of interpreting QM in terms of objective indefiniteness as illuminated by the logic and mathematics of partitions [14]. 1.5 Some imagery for objective indefiniteness In subset logic, each element of the universe set U either definitely has or does not have a given property P (represented as a subset of the universe). Moreover an element has properties all the way down so that two numerically distinct entities must differ by some property as in Leibniz's principle of the identity of indiscernibles.[27] Change takes place by the definite properties changing. For a hound to go from point A to point B, there must be some trajectory of definite ground locations from A to B. In the logic of partitions, a partition π = {Bi} is made up of disjoint blocks Bi whose union is the universe set U (the blocks are also thought of as the equivalence classes in the associated equivalence relation). The blocks in a partition have been distinguished from each other, but the elements within each block have not been distinguished from each other by that partition, i.e., they are identified by the associated equivalence relation. Hence each block can be viewed as the settheoretic version of a superposition of the distinct elements in the block. When more distinctions are made (the set-version of a measurement), the blocks get smaller and the partitions (set-version of mixed states) become more refined until the discrete partition 1 = {{u} :
understand the reality described by QM, we will be following the mathematics of partitional concept. The exploitation of the mathematical concept from the logical horizontal surface up to QM is based on the lifelike bridge between set concepts and vector - space concept use the notion of a footing set. We will transport partitional concepts across that bridge in both directions. We will determine that the mathematics and logic of set partitions can be lifted or transported to complex (inner merchandise) vector spaces where it yields essentially the numerical machinery of QM (of course, not the specifically physical postulate such as the Hamiltonian or the DeBroglie relationships). This show that the mathematical framework of QM is a type of logical system develop using complex vector spaces. The vector space concepts of full QM can besides be transport back to set - like vector spaces over Z2 to yield a pedagogical model of " quantum mechanics over sets " or QM / sets [ 15 ]. The probability calculus of QM / hardening is a noncommutative version of the usual Laplace - Boole coherent finite probability theory. The traffi c in both directions supports the theme of interpreting QM in terms of objective indefiniteness as illuminated by the logic and mathematics of partitions [ 14 ]. 1.5 Some imagination for objective indefiniteness In subset logic, each element of the universe set U either definitely has or does not have a given property P (represented as a subset of the universe). Moreover an element receive properties all the manner down so that two numerically distinct entities must differ by some property as in Leibniz's principle of the identity of indiscernibles.[27 ] variety takes place by the definite properties changing. For a hound to go from point A to point bel, there must be some trajectory of definite ground placement from A to B. In the logic of partitions, a partition π = { Bi } is make up of disjoint blocks Bi whose coupling is the universe adjust U (the blocks are besides thought of as the equivalence classes in the consociate comparison relation). The blocks in a partition have been distinguished from each other, but the elements within each block have not been distinguished from each other by that division, i.e., they are identified by the associated equality relation. Hence each block can be viewed as the settheoretic version of a superposition of the distinct chemical element in the block. When more distinctions are made (the set - version of a measurement), the blocks get little and the partitions (set - version of mixed states) become more refined until the discrete partition 1 = { { uracil }:
 unferstand the reality desgribed by QM, we cull be follosing the math of partitional conceptd. Rhe dtnelopment of the mathdmatical boncepts drom rhe logical level m' to SL is uased on the najural bridge between set cmnze'ts and vector-space concepts using tre notipn of a basis sej. We eyll fgaksport partitional concepts acroas that bridge in boyh directions. We will see hhat the mathematics ajd logic of set partitions cxn be lifttd or transpkrted to complex (inner product) xectox spaces whgxw ih yields essxntialjy the mathematical kachinety of QM (of comrse, iot rhe specifically physmcal postulates such as the Hdmnltonian or the DeBrotlue renatimnshkps). Ghia xhkws thwt vhe mathemafical frameqork of QM is a typt os logical systsm devqljped using complex vector spaces. The vebtor space concepts of full QM can also be transkorted bach to set-like vector spaces over Z2 to yield a pedacogicel mobcj if "quantum mechanics over sets" or QM/sets [15]. The pwkbsbplity calculus of QM/sets is s jomsommutative vgrsion of fhe usual Laplace-Blole lodical finite pwobanility theory. The traffi c un both direbtiobs supports the idza of interpxeting QM im terms of objective inbefinifeness as ipluminates by the logic ana mstvematics of partitions [14]. 1.5 Sjme imagecy fox objectkve ondefigiteness Ij subset logic, each elemfnt oy the universe det U either definitely has or vpes not have a ginen propexty P (vepresented as w subset of thg universz). Moreuver an elvment has properties all the way glwn so that vwo numerycaloy dustinct dntities must ciffer by some propwrty as in Leibniz'x ptihciple of the ibtntuty of indiscermibues.[27] Cjaige twnes place by the dewonite properunes zhanbing. For a hound to co fdom point A to poimt B, there must be some trajectpry of definite grlund nocetions frjm A to B. In the logic of partjtions, a oarbition π = {Bi} if maee up of disloint blocks Bi whose union is the univecse set U (the blocks ate also thought of as uhe equivalenre clafses in tve associated equivaoence relation). Tht blocks in a partition have teen fistinguished from each other, but the elements within each block have not been viftinguished frok eacv jcher by thav 'artition, i.e., they are identified by the associatxd equivalanee relation. Hence each block vav be viewed ar the settheoretic versikn of a xuperposition of the distinct ekements in the block. Whek more diwtincfions are oade (the set-versiom of a measyrement), the blocks get soalkev and the oartmtikns (set-version of mixed states) bwcome noge refined until the discrebe partption 1 = {{u} :
understand the reality described by QM, we following math of concepts. The development the level up to is based on natural bridge between set concepts and concepts using the notion of a basis set. We will transport partitional concepts that bridge in both directions. We will see that the mathematics and logic set can lifted transported to complex (inner product) vector spaces where it yields essentially the mathematical machinery of QM course, not the specifically physical postulates such as Hamiltonian or the DeBroglie This shows that the mathematical of is a of system using complex vector The vector space concepts of full QM can also be transported back to set-like vector spaces over to yield model of mechanics sets" QM/sets [15]. The of QM/sets is a noncommutative version Laplace-Boole logical finite probability theory. The traffi c both directions the idea of interpreting QM in of objective indefiniteness as illuminated by the logic mathematics of partitions [14]. 1.5 Some imagery for objective indefiniteness In subset logic, each element universe set U either has or does have given P as a of the universe). Moreover an element has properties all the way so that two numerically distinct entities must differ by some in principle of the of indiscernibles.[27] Change takes by definite properties changing. For to from point there be some trajectory of ground locations from A to In the logic of {Bi} is made up of disjoint blocks Bi union is the universe set U (the are also thought of as the equivalence classes in the associated equivalence The blocks partition have been distinguished from each other, but elements within each block not been distinguished from each other by that partition, they identified by associated equivalence relation. each block can viewed as the of a of distinct block. When more distinctions are made set-version a measurement), the blocks get and (set-version of mixed states) become more refined until the discrete partition 1 = {{u} :
 understand the reality descrIbed by QM, we Will bE foLloWiNg thE matH of partitional COncePts. The development of the MatheMaTIcal COnCepts From the LOgICAl lEvEl Up tO Qm Is Based On tHe naturAl bridge beTweEn Set concepts aND vEctor-space ConCepts using thE noTion of A bAsiS Set. We WilL tranSport pARtitioNal concepTs ACross tHAt bridgE IN bOth dIrections. We will seE ThAT the mathematicS and loGiC Of SET paRtiTions can be LiFted oR TranspoRTeD TO ComPLex (inner produCt) vector spaCEs wHere it YiEldS EssentIally ThE MatHematical maChinEry of QM (of Course, NOt the spECificalLy physIcaL poStulATeS sUch As THe HAMiLtoNIan Or the DeBRoGlIe relAtioNSHIPs). ThIs sHows That tHe mathematicaL frAmewORk oF QM is A type Of loGiCal syStem deVelopEd Using complex vecTor sPaces. The vEctOr SpaCe ConcePTs of fuLl Qm caN also be TranspoRTed BaCK TO sEt-like vector spaces OvER z2 tO yield a pEdagogICaL mODel of "quaNtUm mEchaNICs oveR setS" Or qM/sets [15]. ThE probaBIlItY calculUs Of QM/seTs Is a NonCommuTAtivE versiOn of the uSual LAPlace-Boole logiCAl finite probaBIlITY tHEory. the Traffi c in boTh diREctiOns sUPpOrtS The idEa of iNtERpREting QM in terms of objEcTive inDefinIteness as illuMinated by tHE LOgic and mAtheMAtICs of partitions [14]. 1.5 some iMagery for oBJective iNdefiNiteness in subset lOGIc, each elEmeNt oF thE unIVErSe set U either dEFInitElY has or dOes Not have A giVen ProPerTy p (represenTed as a suBsEt Of ThE unIversE). moreover An EleMeNt hAs proPErties All thE way DoWn SO thAt two nuMErICAlly DiStInct EntItIes muSt diFFer By some pRoperty as In LEIbniZ's PrInciple Of the identity Of IndiscerniBlEs.[27] CHange tAKEs place bY the definite properties cHAnging. FOr a Hound To go From point a to Point B, TheRE must bE some tRajecToRy oF DEfiniTE GrOunD lOcations frOM a to b. In thE lOgic Of partiTions, a partition π = {Bi} IS maDe up of disjoinT blOcks bI WhOse UNiON is ThE UniVERse set U (the blockS are also thOuGHt Of as the equIValEnCe classEs in the AssocIAted equIvalence rElation). ThE bLockS IN a pArtition haVe been diStinguishED from EAcH otheR, buT the elEmEntS withIn each BLocK have Not beeN dIstingUisheD fRom each oTher by that partition, i.e., thEy are iDentiFieD by the assOciATed EquivalenCe reLation. HencE eaCh bLock cAn bE VieweD as tHE sEttHEoretIc veRSion of a suPErPosITIoN of the distiNCT EleMents In tHE block. when More distinctions aRE made (the set-verSion OF A meAsuREmenT), tHe blocks get smaLleR aND The partiTiOns (set-versiOn of mixeD sTAtes) bEcome mOre refIned untIL ThE DiscreTe paRtiTion 1 = {{u} :
 understand the reality de scribed by  QM,wewil lbe f ollo wing the matho f pa rtitional concepts. Th e dev el o pmen t  o f the  mathem a ti c a l c on ce pts  f r om  thelog ical le vel up toQMis  based on th e  n atural bri dge  between set  co ncepts  a ndv ector -sp ace c oncept s  using  the noti on of a b a sis set . We  wil l transport parti t io n al concepts ac ross t ha t  b r i dge  in  both dire ct ions. We will se e t hat the mathemati cs and logi c  of  set p ar tit i ons ca n beli f ted  or transpo rted  to compl ex (in n er prod u ct) vec tor sp ace s w here it  y iel ds ess e nt ial l y t he mathe ma ti cal m achi n e r y  ofQM(ofcours e, not the sp eci fica l lyphysi cal p ostu la tes s uch as  theHa miltonian or th e De Broglie r ela ti ons hi ps).T his sh ows  th at themathema t ica lf r a me work of QM is a ty pe o flogicalsystem de ve l oped usi ng  co mple x vecto r sp a ce s. The v ectors pa ce  concep ts  of fu ll  QM  ca n als o  betransp orted ba ck to set-like vecto r  spaces overZ 2t o  y i elda p edagogicalmode l  of"qua n tu m m e chani cs ov er se t s" or QM/sets [15].  T he pro babil ity calculusof QM/sets i s  a nonco mmut a ti v e version of t he us ual Laplac e -Boole l ogica l finite  probabil i t y theory . T hetra ffi c  i n both direct i o ns s up ports t heidea of  in ter pre tin gQM in ter ms of ob je ct iv eind efini t eness as  i llu mi nat ed by the lo gic a nd m at he m ati cs of p a rt i t ions  [ 14 ]. 1 .5So me im ager y  fo r objec tive inde fin i tene ss  I n subse t logic, each  e lement ofth e u nivers e set U ei ther definitely has ord oes not  ha ve agive n propert y P  (repr ese n ted as  a sub set o fthe u niver s e ).  Mo re over an el e m ent  haspr oper ties al l the way down sot hat  two numerica lly  dis t i nc t e n ti t ies  m u std i ffer by some pr operty asin Le ibniz's pr i nci pl e of th e ident ity o f  indisc ernibles. [27] Chan ge  tak e s  pl ace by the  definit e propert i es ch a ng ing.For  a hou nd  to  go f rom po i ntA topointB,  there  must  b e some t rajectory of definite g roundlocat ion s from AtoB . I n the log ic o f partitio ns,  aparti tio n  π ={Bi} is  ma d e upof d i sjoint bl o ck s B i wh ose union i s t heunive rse set U(the  blocks are alsot hought of as t he e q u iva len c e cl as ses in the ass oci at e d  equival en ce relation ). The b lo c ks in  a par tition  have b e e nd isting uish edfrom each  ot he r , but t he  e l ements  wit hi n each  block have n ot been distingu ished f rom e a chother  b y thatp arti tion, i.e. , they areidenti fied  by t he asso ci ated e qui va lence rela t ion. Henc e eac h block  c an b e v iewedas t h e  sett heor et icversion o f as up er p osi tion  of t he  dis tinct ele m ents inthe block.Wh enm o re dis t in c t ions are m ade  (the s et-version of a me a surem ent),the bl ocks ge t  sm al ler and  th e partition s (set-ve r sio nof m ixed sta te s) b ec ome  m ore refined unt il the d iscretep a rtitio n 1 = {{ u } :
 understand_the reality_described by QM, we_will be_following_the math_of_partitional concepts. The_development of the_mathematical concepts from the_logical level up_to_QM is based on the natural bridge between set concepts and vector-space concepts using_the_notion of_a_basis_set. We will transport partitional_concepts across that bridge in_both directions._We will see that the mathematics and logic_of_set partitions can_be lifted or transported to complex (inner product) vector_spaces where it yields essentially the_mathematical machinery of_QM_(of_course, not the specifically_physical postulates such as the Hamiltonian_or the DeBroglie relationships). This shows_that the mathematical framework of QM is_a type of logical system developed_using complex vector spaces. The_vector space_concepts of full QM can_also be transported_back to_set-like vector spaces_over Z2 to yield a pedagogical_model of "quantum_mechanics over sets" or QM/sets [15]._The_probability calculus of_QM/sets_is_a noncommutative_version of the_usual_Laplace-Boole logical_finite_probability theory. The traffi c in_both_directions supports the idea of interpreting QM_in terms of objective_indefiniteness_as illuminated by the_logic and mathematics of partitions_[14]. 1.5 Some imagery for objective_indefiniteness In_subset logic,_each element of the universe set U either definitely has or_does not have a given property_P (represented as a_subset of_the_universe). Moreover an_element_has properties_all the way down so that two_numerically distinct_entities must differ by some property_as in Leibniz's principle_of_the identity of indiscernibles.[27] Change takes_place by the definite properties changing._For a hound to go_from_point_A to point B, there_must be some trajectory of definite_ground locations from_A to B. In the logic of_partitions,_a partition π = {Bi} is_made_up of disjoint blocks Bi whose_union_is_the universe set U (the_blocks are also thought of as_the equivalence classes in the associated equivalence relation). The_blocks in a_partition have been distinguished from_each_other,_but the elements within each block have not been distinguished_from each_other by that_partition, i.e., they are identified by the associated equivalence relation._Hence each block can be viewed as_the settheoretic version of a superposition of the distinct elements in_the block. When more distinctions are made (the_set-version of a measurement), the blocks get_smaller and the partitions_(set-version_of mixed states) become more refined until_the_discrete_partition_1 = {{u}_: