-version of a pure state in QM prior to a distinctions-creating measurement that decoheres the pure state into non-blob partitions analogous to a mixed state (see [15] for spelling this out using density matrices). 8This notion of logical in-formation as distinctions is based on partition logic just as logical probability is based on subset logic ([11] and [13]). That is, the logical entropy of a partition is the normalized counting measure of the distinctions of a partition (represented as a binary relation) just as the Laplace-Boole logical probability of a subset is the normalized counting measure on the subsets (events) of the finite universe set (set of equiprobable outcomes). 9 In treating the universe U = {u, u′,...} and the discrete partition 1 = {{u}, {u′},...} as the "same" we are neglecting the distinction between u and {u} for u ∈ U. 9 These two creation stories might also be illustrated as follows. Figure 4: Two creation stories One might think of the universe U (in the middle of the above picture) as the macroscopic world of definite entities that we ordinarily experience. Common sense and classical physics assumes, as it were, the subset creation story on the left. But a priori, it could just as well have been the dual story, the partition creation story pictured on the right, that leads to the same macro-picture U. Since partitions are the mathematical expression of indefiniteness, our strategy is to first show where set partitions come from and then to "lift" or "transport" the partitional machinery to vector spaces using the notion of a basis set of a vector space. The result is essentially the mathematical machinery of quantum mechanics-all of which shows how quantum mechanics can be interpreted using the objective indefiniteness conception of reality that is associated at the logical level with partition logic. 2 Whence set partitions? 2.1 Set partitions from set attributes Take the universe set as some specific set of people, say, in a room. People have numerical attributes like weight, height, or age as well as non-numerical attributes with other values such place of birth, family name, and country of citizenship. Abstractly an attribute on a universe set U is a function f : U → R from U to some set of values R (usually the reals R). In subset logic, an element u ∈ U either has a property represented by a subset S ⊆ U or not; in partition logic
-version of a pure state in QM prior to a distinctions - creating measurement that decoheres the arrant department of state into non - blob partitions analogous to a mixed state (witness [ 15 ] for spelling this out using concentration matrices). 8This notion of logical in - constitution as distinctions is based on division logic just as logical probability is based on subset logic ([ 11 ] and [ 13 ]). That is, the logical information of a partition is the normalized counting measure of the distinctions of a division (represented as a binary relation) just as the Laplace - Boole coherent probability of a subset is the normalized counting measuring stick on the subsets (event) of the finite universe set (set of equiprobable outcomes). 9 In treating the population U = { u, u′, ... } and the discrete partition 1 = { { u }, { u′ }, ... } as the " same " we are neglecting the distinction between uracil and { u } for u ∈ U. 9 These two creation stories might also be illustrated as follows. Figure 4: Two creation stories One might think of the population U (in the center of the above picture) as the macroscopic world of definite entities that we normally have. Common sense and classical physics assumes, as it were, the subset creation story on the left field. But a priori, it could just as well have been the dual story, the partition creation story pictured on the right, that leads to the same macro - picture U. Since partition are the mathematical saying of indefiniteness, our strategy is to first show where set partitions derive from and then to " lift " or " transport " the partitional machinery to vector spaces use the notion of a basis set of a vector space. The solution is essentially the mathematical machinery of quantum mechanism - all of which shows how quantum mechanism can be interpreted practice the objective indefiniteness conception of world that is associated at the logical level with division logic. 2 Whence set partitions? 2.1 Set partitions from set attributes Take the universe set as some specific set of people, say, in a room. People experience numerical attributes like weight unit, height, or age equally well as non - numerical attributes with other values such place of birth, family name, and country of citizenship. Abstractly an attribute on a universe set U is a affair farad: uracil → R from U to some set of values R (usually the reals R). In subset logic, an component u ∈ U either has a property represented by a subset S ⊆ U or not; in division logic
-verdion of a pure state in DM prior to a dnwtinctmons-crezting mexsurement that decoheres the pyre sucte into non-blob pargitions ajalogous to e mixed state (sex [15] for sizllinf thiv out using denxity matriwes). 8This notiot uf logical in-formation as distinctionf is baxef on partition logpc jusf as logical probability is based kn substt logic ([11] and [13]). Thay is, the logical entropy ov a oartition is the nlrmalized ciuntybg measure ow the distpuctions of z partition (represented as a bivary xelation) juwt as jhe Laplace-Biole jogical probability mf a sunset is the novmalived xounting measure on tie subsets (events) of the finide universe set (set od equhprotabld ougcojex). 9 In trfatmng the uniberse U = {u, y′,...} and the discrete karepyion 1 = {{u}, {u′},...} aa the "faie" we are neglecting the distinction beuween u and {u} for u ∈ U. 9 Thesw two creation storied might ajso be illustrated as follows. Figure 4: Two creatiot stocids Inc mieyt think of the universe U (in the middle of the zbpvv picture) as the iacroscopic wlrkq of definite entitnss that we ordinarilj experyence. Common stnse snd classical physics assumws, as it werv, thw subset creation dtory on thz left. But s priori, it could just cs welm have been the dual rtory, the partitkon cseation story pictured on ehe right, thac leads go tne samq macro-pichure M. Since partitions wre tke madhematical expression of indefiniteness, onc strategy is tm fprst show wherc set partitionf come from anb then tj "lifg" or "transiort" the 'artitional iachinery to egctor spaces nsing the notuon if a barks set of a vevtor spacv. Uhe result is essentially thc matfsmatical machinzxy of quantum mecnankcs-wlp pf fhich shows vow duavyum mdchanics cak bd inyerpreted using the mbjedtive indefinitenexs conceptuon of rqality that ix associated at tht logiral letel wiyh kartition logic. 2 Whence set parfitions? 2.1 Det partitions srom set attribuces Take the universe set as some specifmc set of people, say, in a room. People have kumerical atvributqs like waight, height, or age qs well as non-numtrical attributes with kther ealued such place of birth, family name, and country of citizenship. Abstractly ab attrmbtte on a unjverxe sen U ns a fugctili f : U → R from U no some set of values R (usually vhe reals S). Nn subset logic, an element u ∈ U either has a property represented by a subsey S ⊆ U or not; in partition logiv
-version of a pure state in QM a measurement that the pure state a state (see [15] spelling this out density matrices). 8This notion of logical as distinctions is based on partition logic just as logical probability is based subset logic ([11] and [13]). That is, the logical entropy of a partition the counting of distinctions of a partition (represented as a binary relation) just as the Laplace-Boole logical probability of subset is the normalized counting measure on the (events) of the finite set (set of equiprobable outcomes). In the universe = u′,...} the discrete partition = {{u}, {u′},...} as the "same" we are neglecting the distinction between u and {u} for u U. 9 creation stories also illustrated follows. Figure 4: stories One might think of the the middle of the above picture) as the world of entities that we ordinarily experience. Common and classical physics assumes, as it were, the creation story on the left. But a priori, it could just as well have been story, the partition creation pictured on the that to same U. Since are the mathematical expression of indefiniteness, our strategy is to first where set partitions come from and then to "lift" or partitional to vector spaces the notion of a set a vector space. The essentially mathematical mechanics-all which how quantum mechanics can interpreted using the objective indefiniteness of reality that is with partition logic. 2 Whence set partitions? 2.1 partitions from set attributes Take the universe as some specific set of people, say, in a room. People have attributes like or age as well as non-numerical attributes with values such place of family name, and country of citizenship. Abstractly an attribute a set U a function f U → R U to some values R the R). an element u ∈ U either a represented by a subset S U in partition logic
-version of a pure state in QM prIor to a distInctiOns-CreAtIng mEasuRement that decoHEres The pure state into non-bloB partItIOns aNAlOgous To a mixeD StATE (seE [15] fOr SpeLlINg This oUt uSing denSity matricEs). 8THiS notion of logICaL in-formatiOn aS distinctionS is Based oN pArtITion lOgiC just As logiCAl probAbility is BaSEd on suBSet logiC ([11] ANd [13]). that Is, the logical entroPY oF A partition is thE normaLiZEd COUntIng Measure of tHe DistiNCtions oF A pARTItiON (represented aS a binary relATioN) just aS tHe LAPlace-BOole lOgICal Probability Of a sUbset is thE normaLIzed couNTing meaSure on The SubSets (EVeNtS) of ThE FinITe UniVErsE set (set oF eQuIprobAble OUTCOmes). 9 in tReatIng thE universe U = {u, u′,...} aNd tHe diSCreTe parTitioN 1 = {{u}, {u′},...} aS tHe "samE" we are NegleCtIng the distinctiOn beTween u and {U} foR u ∈ u. 9 ThEsE two cREation StoRieS might aLso be ilLUstRaTED As Follows. Figure 4: Two crEaTIOn Stories ONe mighT ThInK Of the uniVeRse u (in tHE MiddlE of tHE aBove pictUre) as tHE mAcRoscopiC wOrld of DeFinIte EntitIEs thAt we orDinarily ExperIEnce. Common sensE And classical pHYsICS aSSumeS, as It were, the suBset CReatIon sTOrY on THe lefT. But a PrIOrI, It could just as well haVe Been thE dual Story, the partiTion creatiON STory pictUred ON tHE right, that leadS to thE same macro-PIcture U. SInce pArtitionS are the maTHEmatical ExpResSioN of INDeFiniteness, our STRateGy Is to firSt sHow wherE seT paRtiTioNs Come from aNd then to "LiFt" Or "TrAnsPort" tHE partitiOnAl mAcHinEry to VEctor sPaces UsinG tHe NOtiOn of a baSIs SET of a VeCtOr spAce. thE resuLt is ESseNtially The mathemAtiCAl maChInEry of quAntum mechanicS-aLl of which sHoWs hOw quanTUM mechaniCs can be interpreted using THe objecTivE indeFiniTeness conCepTion of ReaLIty thaT is assOciatEd At tHE LogicAL LeVel WiTh partitioN LOgiC. 2 WhenCe Set pArtitioNs? 2.1 Set partitions froM Set Attributes TakE thE uniVERsE seT As SOme SpECifIC Set of people, say, iN a room. PeopLe HAvE numerical ATtrIbUtes likE weight, HeighT, Or age as Well as non-Numerical AtTribUTEs wIth other vaLues such Place of biRTh, famILy Name, aNd cOuntry Of CitIzensHip. AbsTRacTly an AttribUtE on a unIversE sEt U is a fuNction f : U → R from U to some set Of valuEs R (usUalLy the realS R). IN SubSet logic, aN eleMent u ∈ U eithEr hAs a PropeRty REpresEnteD By A suBSet S ⊆ U Or noT; In partitiON lOgiC
-version of a pure state i n QM prior  to a  di sti nc tion s-cr eating measure m entthat decoheres the pur e sta te into no n-blo b parti t io n s  an al og ous  t o  a  mixe d s tate (s ee [15] fo r s pe lling this o u tusing dens ity  matrices).8Th is not io n o f  logi cal  in-f ormati o n as d istinctio ns is bas e d on pa r t it ionlogic just as log i ca l  probability i s base do ns u bse t l ogic ([11]  a nd [1 3 ]). Tha t  i s , the logical entro py of a par t iti on isth e n o rmaliz ed co un t ing  measure of  the  distinct ions o f  a part i tion (r eprese nte d a s ab in ar y r el a tio n )jus t  as  the Lap la ce -Bool e lo g i c a l pr oba bili ty of  a subset isthe  nor m ali zed c ounti ng m ea sureon the  subs et s (events) of t he f inite uni ver se  se t(seto f equi pro bab le outc omes).9  In  t r e a ti ng the universe U={ u ,u′,...}and th e  d is c rete par ti tio n 1= {{u},  {u′ } ,. ..} as t he "sa m e"  w e are n eg lectin gthe  di stinc t ionbetwee n u and{u} f o r u ∈ U. 9 The s e two creatio n  s t o ri e s mi ght  also be il lust r ated  asf ol low s . Fig ure 4 :T wo creation stories On emightthink  of the unive rse U (int h e  middleof t h ea bove picture)as th e macrosco p ic world  of d efiniteentitiest h at we or din ari lyexp e r ie nce. Common s e n se a nd  classi cal  physic s a ssu mes , a sit were,the subs et  c re at ion  stor y  on thele ft.  B uta pri o ri, it  coul d ju st  a s  we ll have be e n  the  d ua l st ory ,the p arti t ion  creati on storypic t ured  o nthe rig ht, that lead sto the sam emac ro-pic t u re U. Si nce partitions are them athemat ica l exp ress ion of in def initen ess ,  our s trateg y isto  fi r s t sho w wh ere  s et partiti o n s c ome f ro m an d thento "lift" or "tran s por t" the partit ion al m a c hi ner y  t o  ve ct o r s p a ces using the n otion of a  b a si s set of a vec to r space . The r esult is esse ntially t he mathem at ical m ach inery of q uantum m echanics- a ll of wh ich s how s howqu ant um me chanic s  ca n beinterp re ted us ing t he  objecti ve indefiniteness conce ptionof re ali ty that i s a s soc iated atthelogical le vel  wi th pa rti t ion l ogic .  2  Wh e nce s et p a rtitions? 2. 1 S e t  p artitions f r o m  se t att rib u tes Ta ke t he universe set a s  some specific  set o f p eop l e, s ay , in a room. P eop le h ave nume ri cal attribu tes like  w e ight,  heigh t, orage asw e ll as non -num eri cal attri but es with ot he rv aluessuch  p lace o f birt h , fa m i ly name, and cou ntryo f  citi z ens hip.Ab stractl y  anattributeon a univer se set  U i s a f unction  f  : U →  Rfr om U to so m e set ofvalue s R (us ua llythe  reals  R). I n sub setlo gic , an elem e n tu  ∈  U eit herhas a  p rope rty repre s ented by  as ubset S  ⊆  Uo r  not;i np a rtition lo gic 
-version of_a pure_state in QM prior_to a_distinctions-creating_measurement that_decoheres_the pure state_into non-blob partitions_analogous to a mixed_state (see [15]_for_spelling this out using density matrices). 8This notion of logical in-formation as distinctions is_based_on partition_logic_just_as logical probability is based_on subset logic ([11] and_[13]). That_is, the logical entropy of a partition is_the_normalized counting measure_of the distinctions of a partition (represented as a_binary relation) just as the Laplace-Boole_logical probability of_a_subset_is the normalized counting_measure on the subsets (events) of_the finite universe set (set of_equiprobable outcomes). 9 In treating the universe_U = {u, u′,...} and the_discrete partition 1 = {{u},_{u′},...} as_the "same" we are neglecting_the distinction between_u and_{u} for u_∈ U. 9 These two creation_stories might also_be illustrated as follows. Figure 4:_Two_creation stories One_might_think_of the_universe U (in_the_middle of_the_above picture) as the macroscopic world_of_definite entities that we ordinarily experience. Common_sense and classical physics_assumes,_as it were, the_subset creation story on the_left. But a priori, it could_just as_well have_been the dual story, the partition creation story pictured on the_right, that leads to the same_macro-picture U. Since partitions_are the_mathematical_expression of indefiniteness,_our_strategy is_to first show where set partitions come_from and_then to "lift" or "transport" the_partitional machinery to vector_spaces_using the notion of a basis_set of a vector space. The_result is essentially the mathematical_machinery_of_quantum mechanics-all of which shows_how quantum mechanics can be interpreted_using the objective_indefiniteness conception of reality that is associated_at_the logical level with partition logic._2_Whence set partitions? 2.1 Set partitions_from_set_attributes Take the universe set_as some specific set of people,_say, in a room. People have numerical attributes like_weight, height, or_age as well as non-numerical_attributes_with_other values such place of birth, family name, and country_of citizenship._Abstractly an attribute_on a universe set U is a function f :_U → R from U to some_set of values R (usually the reals R). In subset logic,_an element u ∈ U either has a_property represented by a subset S ⊆_U or not; in_partition_logic