 the join of compatible vector-space partitions-where one partition describes the distinction-making measurement apparatus representing the observable being measured and the other partition is the indiscrete (i.e., one block) partition representing the given pure state to be measured or, better, to be decohered.28 Using the respective partition join operations, a complete set of compatible attributes (CSCA) gives a non-degenerate measurement in QM/sets, and a complete set of commuting operators (CSCO) gives a non-degenerate measurement in full QM. Thus the clear and distinct distinction between "Type 1" distinction-making and "Type 2" distinction-preserving processes in the pedagogical model of QM/sets makes sense of the 26Thus the gates in quantum computing over Z2 are the non-singular linear transformations ([35] and [15]). 27See the previous Figure 3 on how "substance" is constant or conserved in the partition lattice regardless of whether the changes preserve distinctions ("Type 2" processes-imaged in Figure 8 as lateral moves in the lattice) or create distinctions ("Type 1" processes-imaged as upward moves in the lattice). Also the "Type 1" processes that create distinctions will destroy or wipe out distinctions in another incompatible basis. 28See [15] for the development of this using density matrices. This partition mathematical description of the Type 1 measurement process does not give the physical characterization of a distinction-creating measurement apparatus anymore than this approach gives the physical characterization of the distinction-preserving Type 2 unitary processes. 25 von Neumann Type 1 distinction-making measurements and Type 2 distinction-preserving unitary transformations in full QM. 6 Final remarks There are two meta-physical visions of reality suggested by classical physics (objectively definite reality) and by quantum physics (objectively indefinite reality). The problem of interpreting QM is essentially the problem of making sense out of the notion of objective indefiniteness. Our sensemaking strategy was to follow the lead of the mathematics. The definiteness of classical physics is associated with the notion of a subset and is logically expressed in the classical Boolean logic of subsets. The indefiniteness of quantum physics is associated with the notion of a quotient set, equivalence relation, or partition, and the corresponding logic is the recently developed logic of partitions [12]. Moreover, those associated notions of subsets and quotient sets are category-theoretically dual to one another, so from that viewpoint, those are the only two possible frameworks to describe reality. Common sense and classical
the join of compatible vector - space partitions - where one partition trace the differentiation - making measurement apparatus representing the observable being quantify and the other partition is the indiscrete (i.e., one auction block) division representing the given pure state to be measured or, good, to be decohered.28 Using the respective division join operations, a complete set of compatible attributes (CSCA) gives a non - degenerate measurement in QM / hardening, and a complete set of commuting hustler (CSCO) gives a non - degenerate measurement in full QM. Thus the clean and distinct distinction between " Type 1 " distinction - making and " character 2 " distinction - preserving processes in the pedagogical model of QM / sets makes common sense of the 26Thus the gate in quantum computing over Z2 are the non - singular linear transformations ([ 35 ] and [ 15 ]). 27See the previous Figure 3 on how " substance " is constant or conserved in the division wicket regardless of whether the changes preserve distinctions (" character 2 " summons - imaged in Figure 8 as lateral moves in the lattice) or create distinctions (" Type 1 " processes - image as upward moves in the lattice). Also the " Type 1 " processes that create eminence will destroy or wipe out distinctions in another incompatible basis. 28See [ 15 ] for the growth of this using concentration matrix. This partition mathematical description of the Type 1 measurement process does not give the physical characterization of a distinction - create measurement apparatus anymore than this approach gives the physical characterization of the eminence - preserving Type 2 one processes. 25 von Neumann Type 1 distinction - making measurements and Type 2 differentiation - preserving one transformations in full QM. 6 concluding remarks There are two meta - physical visions of reality propose by classical purgative (objectively definite reality) and by quantum physics (objectively indefinite reality). The problem of interpret QM is essentially the problem of making common sense out of the notion of objective indefiniteness. Our sensemaking strategy was to follow the lead of the mathematics. The definiteness of classical physics is consociate with the notion of a subset and is logically expressed in the classical Boolean logic of subsets. The indefiniteness of quantum physics is associated with the notion of a quotient hardening, equality relation, or partition, and the corresponding logic is the recently developed logic of partitions [ 12 ]. Moreover, those associate notions of subsets and quotient sets are class - theoretically dual to one another, therefore from that viewpoint, those are the only two possible model to describe reality. Common sense and classical
 thf join of compatible vecuor-space partitiouw-where one pzrtition describes the distinction-manibg meqsurement apparatus reoresentinh the obwerveble being measucsd and bke otgcr paxtmtion is the inciscrete (i.a., one block) pastktnon representing the given pure statq to be mfasured or, betjer, tp be svcihered.28 Using the respective pzrtitioi join operatioms, a complete set of compahiblf attributes (CSCA) hives a non-eegegwrate measurdment in QM/sets, and a domplete set of commuting operagors (ESCO) gives q bon-fggenerate meesuremvnt in full QM. Thus tve cleat and distinct divtibction between "Type 1" vistinction-making anq "Type 2" dhscinction-preserving pricwsses in dhe owdaeogjcel jodel lf SM/sets makss sense of the 26Thus the gates ig quantum comphting jvqr Z2 are the non-singular linear transfogmatjons ([35] and [15]). 27See the prevuous Figure 3 on how "sobstance" if constant or conserved in the partition lattice segarvldss on whdrhfr the changes preserve distinctions ("Type 2" prjdexsvs-imaged in Figurc 8 as lateral movex ln jhe lattice) or creatz djstinctions ("Type 1" orocessgs-imagwd as upwwrd koves in the lattice). Also tye "Type 1" probessws that create discinctions wiul dgstroy or wipe out distinctious in znother inclmpatible casis. 28See [15] for tfe ceeelopmenu of this using degsity matcices. This pxrtijion maehematical descvhption of the Type 1 meadusement profess does not give the physical characterizatoot ox a distnnctiok-creating measuwement apparatos anymorz than this apprkach gites the physycal characteslzation of tie distinstiob-prewerving Gype 2 unitary lrocesses. 25 von Neumqnn Type 1 distinctlon-mayjng measurementw abd Type 2 distinvtiun-pwedecving gnitary tranvforoatkpns iv full QM. 6 Ninxl rrmarks There are two metz-physical visions pf reality suggestqd by classicsl physics (objectigely vefinive reakiti) and by quantum physics (objectjvely indffikite reality). Ehe irobjem of intzrpreting QM is essentially the problem if making sense oot of the notion of oyjtctive indefiiitenefs. Our setsemaking strategy wqs to follow the kead of the mathematids. The defijiteness of classical physics is associated with the notion of a subset abd is logically exlresxed it che classycal Uoolean logic of xubsets. The indefiniteness of qnantum phyvies is associated with the notoov of a quotievt set, equivalence relatjon, or psrtition, and the corresponding kogic is the recently deyeloped ligic kf parnjtmons [12]. Moreover, those assocnated botions of dubsets and qjotoekt sets arf cauegoxy-theoretically dual to one anothwr, so drlm that viewpoinv, those are bht only jwo possyble frameworks fo dsdcribg reality. Commpn sense and ckassical
the join of compatible vector-space partitions-where one the measurement apparatus the observable being is indiscrete (i.e., one partition representing the pure state to be measured or, to be decohered.28 Using the respective partition join operations, a complete set of attributes (CSCA) gives a non-degenerate measurement in QM/sets, and a complete set of operators gives non-degenerate in full QM. Thus the clear and distinct distinction between "Type 1" distinction-making and "Type 2" processes in the pedagogical model of QM/sets makes of the 26Thus the in quantum computing over Z2 the linear transformations and 27See previous Figure 3 how "substance" is constant or conserved in the partition lattice regardless of whether the changes preserve distinctions 2" processes-imaged 8 as moves the or create distinctions processes-imaged as upward moves in the "Type 1" processes that create distinctions will destroy wipe out in another incompatible basis. 28See [15] the development of this using density matrices. This mathematical description of the Type 1 measurement process does not give the physical characterization of measurement apparatus anymore than approach gives the characterization the Type unitary processes. von Neumann Type 1 distinction-making measurements and Type 2 distinction-preserving unitary in full QM. 6 Final remarks There are two meta-physical reality by classical physics definite reality) and by physics indefinite reality). The problem QM essentially making out the notion of objective Our sensemaking strategy was to the lead of the physics is associated with the notion of a and is logically expressed in the classical logic of subsets. The indefiniteness of quantum physics is associated with the of a equivalence relation, or partition, and the corresponding logic the recently developed logic partitions [12]. Moreover, those associated notions of subsets and sets category-theoretically dual one another, so that viewpoint, those the only two to describe Common and
 the join of compatible vector-Space partiTions-WheRe oNe PartItioN describes the dIStinCtion-making measurement ApparAtUS repREsEntinG the obsERvABLe bEiNg MeaSuREd And thE otHer partItion is the IndIsCrete (i.e., one blOCk) Partition rEprEsenting the gIveN pure sTaTe tO Be meaSurEd or, bEtter, tO Be decoHered.28 UsinG tHE respeCTive parTITiOn joIn operations, a compLEtE Set of compatiblE attriBuTEs (cscA) gIveS a non-degenErAte meASuremenT In qm/SEts, ANd a complete seT of commutinG OpeRators (cScO) gIVes a noN-degeNeRAte Measurement In fuLl QM. Thus tHe cleaR And distINct distInctioN beTweEn "TyPE 1" dIsTinCtIOn-mAKiNg aND "TyPe 2" distinCtIoN-presErviNG PROcesSes In thE pedaGogical model oF QM/Sets MAkeS sensE of thE 26ThuS tHe gatEs in quAntum CoMputing over Z2 are The nOn-singulaR liNeAr tRaNsforMAtions ([35] And [15]). 27see The prevIous FigURe 3 oN hOW "SUbStance" is constant or CoNSErVed in the PartitIOn LaTTice regaRdLesS of wHETher tHe chANgEs preserVe distINcTiOns ("Type 2" PrOcesseS-iMagEd iN FiguRE 8 as lAteral Moves in tHe latTIce) or create disTInctions ("Type 1" pROcESSeS-ImagEd aS upward moveS in tHE latTice). aLsO thE "type 1" pRocesSeS ThAT create distinctions WiLl destRoy or Wipe out distinCtions in anOTHEr incompAtibLE bASis. 28See [15] for the deVelopMent of this USing densIty maTrices. ThIs partitiON MathematIcaL deScrIptION oF the Type 1 measuREMent PrOcess doEs nOt give tHe pHysIcaL chArActerizatIon of a diStInCtIoN-crEatinG MeasuremEnT apPaRatUs anyMOre thaN this ApprOaCh GIveS the phySIcAL CharAcTeRizaTioN oF the dIstiNCtiOn-preseRving Type 2 UniTAry pRoCeSses. 25 von neumann Type 1 diStInction-makInG meAsuremENTs and TypE 2 distinction-preserving uNItary trAnsFormaTionS in full QM. 6 finAl remaRks tHere arE two meTa-phySiCal VISions OF ReAliTy Suggested bY CLasSical PhYsicS (objectIvely definite realiTY) anD by quantum phySicS (objECTiVelY InDEfiNiTE reALIty). The problem of InterpretiNg qm iS essentialLY thE pRoblem oF making Sense OUt of the Notion of oBjective iNdEfinITEneSs. Our senseMaking stRategy was TO follOW tHe leaD of The matHeMatIcs. ThE definITenEss of ClassiCaL physiCs is aSsOciated wIth the notion of a subset anD is logIcallY exPressed in The CLasSical BoolEan lOgic of subsEts. the IndefIniTEness Of quANtUm pHYsics Is asSOciated wiTH tHe nOTIoN of a quotienT SET, eqUivalEncE RelatiOn, or Partition, and the coRResponding logiC is tHE RecEntLY devElOped logic of parTitIoNS [12]. moreover, ThOse associatEd notionS oF SubseTs and qUotienT sets arE CAtEGory-thEoreTicAlly dual tO onE aNOther, so FrOm THat vieWpoiNt, Those aRe the oNLy twO POssible frameworkS to deSCRibe rEAliTy. ComMoN sense aND claSsical
 the join of compatible ve ctor-space  part iti ons -w here  one  partition des c ribe s the distinction-maki ng me as u reme n tappar atus re p re s e nti ng  t heob s er vable  be ing mea sured andthe  o ther partiti o nis the ind isc rete (i.e.,one  block )par t ition  re prese ntingt he giv en pure s ta t e to b e  measur e d  o r, b etter, to be deco h er e d.28 Using the  respe ct i ve p art iti on join op er ation s , a com p le t e set of compatible  attributes (CS CA) gi ve s a non-de gener at e  me asurement i n QM /sets, an d a co m plete s e t of co mmutin g o per ator s  ( CS CO)  g i ves anon - deg enerateme as ureme nt i n f u ll Q M.Thus  theclear and dis tin ct d i sti nctio n bet ween  " Type1" dis tinct io n-making and "T ype2" distin cti on -pr es ervin g  proce sse s i n the p edagogi c almo d e l  o f QM/sets makes se ns e of  the 26T hus th e  g at e s in qua nt umcomp u t ing o verZ 2are thenon-si n gu la r linea rtransf or mat ion s ([3 5 ] an d [15] ). 27See  thep revious Figure 3 on how "sub s ta n c e" is c ons tant or con serv e d in  the pa rti t ion l attic er eg a rdless of whether t he  chang es pr eserve distin ctions ("T y p e  2" proc esse s -i m aged in Figure  8 as  lateral m o ves in t he la ttice) o r created i stinctio ns("T ype  1" p ro cesses-imaged a s up wa rd move s i n the l att ice ).Als othe "Type  1" proc es se sth atcreat e  distinc ti ons  w ill  dest r oy orwipeoutdi st i nct ions in an o t herin co mpat ibl ebasis . 28 S ee[15] fo r the dev elo p ment  o fthis us ing density m at rices. Thi spar tition m athemati cal description of theT ype 1 m eas ureme nt p rocess do esnot gi vet he phy sicalchara ct eri z a tiono f  a  di st inction-cr e a tin g mea su reme nt appa ratus anymore than thi s approach gi ves  the p hy sic a lc har ac t eri z a tion of the dis tinction-p re s er ving Type2  un it ary pro cesses.  25 v o n Neuma nn Type 1  distinct io n-ma k i ngmeasuremen ts and T ype 2 dis t incti o n- prese rvi ng uni ta rytrans format i ons  in f ull QM .6 Fina l rem ar ks There  are two meta-physicalvision s ofrea lity sugg est e d b y classic al p hysics (ob jec tiv ely d efi n ite r eali t y)  an d  by q uant u m physics (o bje c t iv ely indefin i t e  re ality ).T he pro blem  of interpretingQ M is essential ly t h e  pr obl e m of  m aking sense ou t o ft h e notion  o f objective  indefin it e ness.  Our s ensema king st r a te g y wasto f oll ow the le adof the mat he ma t ics. T he d ef initen ess of clas s i cal physics is a ssoci a t ed wi t h t he no ti on of a subs et and islogically e xpress ed i n the  classi ca l Bool ean  l ogic of su b sets. The  inde finiten es s of  qu antumphys i c s isasso ci ate d with th e no t io no f a  quo tient  s et,equivalen c e relati on, or part it ion , and th e  c o r responding  lo gic i s the recent l y de v el o ped l ogic o f part itions[ 12] .Moreove r,t h ose assoc iated not i ons  o f su bsets an dquot ie ntse ts are category -theoret ically d u a l to o nea noth e r, sofr om that view point ,t hose ar e the  only t w o p oss ibl e fr ame work s  to describe reality. Co mmon  sens e  an d cla ssica l
 the_join of_compatible vector-space partitions-where one_partition describes_the_distinction-making measurement_apparatus_representing the observable_being measured and_the other partition is_the indiscrete (i.e.,_one_block) partition representing the given pure state to be measured or, better, to be_decohered.28_Using the_respective_partition_join operations, a complete set_of compatible attributes (CSCA) gives_a non-degenerate_measurement in QM/sets, and a complete set of_commuting_operators (CSCO) gives_a non-degenerate measurement in full QM. Thus the clear_and distinct distinction between "Type 1"_distinction-making and "Type_2"_distinction-preserving_processes in the pedagogical_model of QM/sets makes sense of_the 26Thus the gates in quantum_computing over Z2 are the non-singular linear_transformations ([35] and [15]). 27See the_previous Figure 3 on how_"substance" is_constant or conserved in the_partition lattice regardless_of whether_the changes preserve_distinctions ("Type 2" processes-imaged in Figure_8 as lateral_moves in the lattice) or create_distinctions_("Type 1" processes-imaged_as_upward_moves in_the lattice). Also_the_"Type 1"_processes_that create distinctions will destroy or_wipe_out distinctions in another incompatible basis. 28See_[15] for the development_of_this using density matrices._This partition mathematical description of_the Type 1 measurement process does_not give_the physical_characterization of a distinction-creating measurement apparatus anymore than this approach gives_the physical characterization of the distinction-preserving_Type 2 unitary processes._25 von_Neumann_Type 1 distinction-making_measurements_and Type_2 distinction-preserving unitary transformations in full QM._6 Final_remarks There are two meta-physical visions_of reality suggested by_classical_physics (objectively definite reality) and by_quantum physics (objectively indefinite reality). The_problem of interpreting QM is_essentially_the_problem of making sense out_of the notion of objective indefiniteness._Our sensemaking strategy_was to follow the lead of the_mathematics._The definiteness of classical physics is_associated_with the notion of a subset_and_is_logically expressed in the classical_Boolean logic of subsets. The indefiniteness_of quantum physics is associated with the notion of_a quotient set,_equivalence relation, or partition, and_the_corresponding_logic is the recently developed logic of partitions [12]. Moreover,_those associated_notions of subsets_and quotient sets are category-theoretically dual to one another, so_from that viewpoint, those are the only_two possible frameworks to describe reality. Common sense and classical