 subsets and partitions are thus associated with two aforementioned metaphysical types of reality: 1. the common-sense notion of objectively definite reality assumed in classical physics, and 2. the notion of objectively indefinite reality suggested by quantum mechanics. Subsets and quotient sets (or partitions) are mathematically dual concepts in the reverse-thearrows sense of category-theoretic duality, e.g., a subset is the direct image of a set monomorphism while a set partition is the inverse image of an epimorphism. This duality is familiar in abstract algebra in the interplay of subobjects (e.g., subgroups, subrings, etc.) and quotient objects. William Lawvere calls the general category-theoretic notion of a subobject a part, and then he notes: "The dual notion (obtained by reversing the arrows) of 'part'is the notion of partition."[28, p. 85] The logic appropriate for the usual notion of fully definite reality described by subsets is the ordinary Boolean logic of subsets [3] (usually mis-specified as the special case of "propositional" logic). We have seen that the other vision of objectively indefinite reality suggested by QM is mathematically described by the equivalent concepts of quotient sets, partitions, or equivalence relations. The Boolean logic of subsets has an equally fundamental logic of the dual quotient sets, equivalence relations, or partitions ([12] and [16]).1 The two logics of the dual concepts are associated with the two visions of reality. Boole developed not only the logic of subsets, he defined a normalized counting measure on subsets of a finite universe set to yield the notion of logical finite probability theory. By making the 1The Boolean logic of subsets and the logic of partitions are equally fundamental in that they are the two logics that take (subsets of and partitions on) arbitrary universe sets in their semantics. Other logics that have a precise semantics, such as intuitionistic logic, have some additional structures, such as topologies, order relations, accessibility relations, etc. on the universe sets and thus those logics partake of the specific nature of those additional structures. 5 same mathematical moves in the logic of partitions, i.e., by defining a normalized counting measure on the partitions on a finite universe set (where partitions are represented as the complement of the associated equivalence relations), one obtains a notion of logical entropy ([11] and [13]) that adds to the interpretation of QM mathematics.2 Since our topic is to better understand objective indefiniteness, and thus to better
subsets and partitions are thus associated with two aforementioned metaphysical type of world: 1. the common - sense impression of objectively definite reality assume in classical physics, and 2. the impression of objectively indefinite world suggested by quantum mechanics. Subsets and quotient sets (or partitions) are mathematically double concepts in the reverse - thearrows common sense of category - theoretic duality, e.g., a subset is the direct image of a set monomorphism while a fit partition is the inverse image of an epimorphism. This duality is conversant in abstract algebra in the interplay of subobjects (e.g., subgroups, subrings, etc .) and quotient object. William Lawvere calls the cosmopolitan category - theoretical notion of a subobject a part, and then he notes: " The dual impression (obtained by reversing the arrows) of' part'is the notion of partition. "[28, p. 85 ] The logic appropriate for the usual notion of fully definite reality described by subsets is the ordinary Boolean logic of subsets [ 3 ] (usually mis - specified as the special case of " propositional " logic). We have watch that the other vision of objectively indefinite reality suggest by QM is mathematically described by the equivalent concepts of quotient sets, partition, or equivalence relations. The Boolean logic of subsets has an evenly fundamental logic of the dual quotient sets, equivalence relations, or partitions ([ 12 ] and [ 16]).1 The two logics of the dual concepts are associated with the two visions of reality. Boole develop not only the logic of subset, he specify a normalized counting measure on subsets of a finite universe set to yield the impression of logical finite probability hypothesis. By making the 1The Boolean logic of subsets and the logic of partitions are equally fundamental in that they are the two logic that take (subset of and partitions on) arbitrary universe sets in their semantics. Other logic that have a accurate semantics, such as intuitionistic logic, have some additional structures, such as topology, order relation, accessibility sexual intercourse, etc. on the universe sets and thus those logics partake of the specific nature of those additional structure. 5 same mathematical move in the logic of partitions, i.e., by defining a normalized counting measure on the partitions on a finite population set (where partitions are represented as the complement of the associated equivalence relations), one obtains a notion of logical entropy ([ 11 ] and [ 13 ]) that lend to the rendition of QM mathematics.2 Since our subject is to better understand objective indefiniteness, and therefore to better
 suhsets and partitions are thus associateb with vwo afodementioved metaphysical types of reelitt: 1. tht common-sense notiov of objebtively dwfinmte reality assumed in classiczp phvsmcs, and 2. the nojion of objewtively indefitige reality suggested by quantum mechagics. Sunsfts and quotiegt stts (or lartitions) are mathematically dual concepus in the reverse-tnearrows sense of category-hheogetic duality, e.g., a subset is jge qurect image uf a set monomorphism shile a set partition is the inxerse image of qn eplkorphism. Thms duajity is familiar in dbstracy algebra in tme inverpoay of subobjects (e.g., vubgroups, subrings, etc.) and xuktient objects. Wioluam Ldwvese cxols tht gxnedal cahegkry-theoretjc notion od a subobject a paru, age then he notss: "The dtal notion (obtained by reversing the argows) of 'part'is the notion od partition."[28, p. 85] The lohic appro[riate for the usual notion of fully definite reanity vercrnncd bh skbsets is the ordinary Boolean logic of subseea [3] (uxually mis-speclfied as the specisl csfe of "proposijional" logjc). We have seen thwt the jther vision os obkectively indefinite realitt suggested yy WM is mathematicalpy describeb by tne eqiivalent concepts of quutieht sets, parhitions, od equivalence relxtipnv. The Boolean logic of subfets has en eqbally fuvdamgntal ljgic of thf dual quotient sets, equlvaleuce ralations, og partitions ([12] and [16]).1 The two logirx of the duak wonbepts are assogiated with the two visions oy realitv. Boold developes not oily the logis of subsets, vg defined a nmrmalizeq coyntibg measjfe on subsets pf a finine unicerse set to yield thg hotion of logiccu finite probabikith trelrb. By idking the 1Tha Boulexm logkc of subsebs xnd yhe logic of partitimns zre equally fundamrnbal in thqt they wre the two lpgics that take (suhsets of and larjitions on) arbitrary universe ssts in thfir semantics. Oeher logics that have a precise semantics, such as intuivionistic logic, have sone additional strucjurts, such as to'ologiqs, order selations, accessibiluty relations, etc. on the universe sets and tvus tjose logics partake of the specific nature of those additional structures. 5 same mathematicam mofes it che logic of 'actitions, i.e., by degining a normalized counting meesure on tve partitions on a finite univrrre set (where kartitions are represented as the vomplement of the associated eqiivalence relations), one pbtains q notjon of logmcal entropy ([11] and [13]) tnat adbs to rhe intwrprctation of QM oatnelatics.2 Sijce pux topic is to better understand ovjectice indefiniteness, end thus to neuter
subsets and partitions are thus associated with metaphysical of reality: the common-sense notion in physics, and 2. notion of objectively reality suggested by quantum mechanics. Subsets quotient sets (or partitions) are mathematically dual concepts in the reverse-thearrows sense of duality, e.g., a subset is the direct image of a set monomorphism while set is inverse of an epimorphism. This duality is familiar in abstract algebra in the interplay of subobjects (e.g., subrings, etc.) and quotient objects. William Lawvere calls general category-theoretic notion of subobject a part, and then notes: dual notion by the of 'part'is the of partition."[28, p. 85] The logic appropriate for the usual notion of fully definite reality described by is the logic of [3] mis-specified the special case logic). We have seen that the objectively indefinite reality suggested by QM is mathematically by the concepts of quotient sets, partitions, or relations. The Boolean logic of subsets has an fundamental logic of the dual quotient sets, equivalence relations, or partitions ([12] and [16]).1 The of the dual concepts associated with the visions reality. developed only the of subsets, he defined a normalized counting measure on subsets of finite universe set to yield the notion of logical finite By the 1The Boolean of subsets and the of are equally fundamental in are two (subsets and on) arbitrary universe sets their semantics. Other logics that a precise semantics, such additional structures, such as topologies, order relations, accessibility etc. on the universe sets and thus logics partake of the specific nature of those additional structures. 5 same moves in of partitions, i.e., by defining a normalized counting on the partitions on finite universe set (where partitions are represented as the of associated equivalence one obtains a of logical entropy and [13]) that the interpretation QM Since to better understand objective indefiniteness, and to
 subsets and partitions are thUs associatEd witH twO afOrEmenTionEd metaphysical TYpes Of reality: 1. the common-sensE notiOn OF objECtIvely DefinitE ReALIty AsSuMed In CLaSsicaL phYsics, anD 2. the notion Of oBjEctively indeFInIte reality SugGested by quanTum MechanIcS. SuBSets aNd qUotieNt sets (OR partiTions) are mAtHEmaticALly dual CONcEpts In the reverse-thearROwS Sense of categorY-theorEtIC dUALitY, e.g., A subset is tHe DirecT Image of A SeT MONomORphism while a sEt partition IS thE inverSe ImaGE of an ePimorPhISm. THis duality iS famIliar in abStract ALgebra iN The inteRplay oF suBobJectS (E.g., SuBgrOuPS, suBRiNgs, ETc.) aNd quotieNt ObJects. willIAM lAwveRe cAlls The geNeral category-TheOretIC noTion oF a subObjeCt A part, And theN he noTeS: "The dual notion (oBtaiNed by reveRsiNg The ArRows) oF 'Part'is The NotIon of paRtition."[28, P. 85] the LoGIC ApPropriate for the usuAl NOTiOn of fullY definITe ReALity descRiBed By suBSEts is The oRDiNary BoolEan logIC oF sUbsets [3] (uSuAlly miS-sPecIfiEd as tHE speCial caSe of "propOsitiONal" logic). We have SEen that the othER vISIoN Of obJecTively indefInitE RealIty sUGgEstED by QM Is matHeMAtICally described by the EqUivaleNt conCepts of quotieNt sets, partITIOns, or equIvalENcE Relations. The BoOlean Logic of subSEts has an EqualLy fundamEntal logiC OF the dual QuoTieNt sEts, EQUiValence relatiONS, or pArTitions ([12] And [16]).1 the two lOgiCs oF thE duAl Concepts aRe associAtEd WiTh The Two viSIons of reAlIty. boOle DevelOPed not Only tHe loGiC oF SubSets, he dEFiNED a noRmAlIzed CouNtIng meAsurE On sUbsets oF a finite uNivERse sEt To Yield thE notion of logiCaL finite proBaBilIty theORY. By makinG the 1The Boolean logic of suBSets and The Logic Of paRtitions aRe eQually FunDAmentaL in thaT they ArE thE TWo logICS tHat TaKe (subsets oF ANd pArtitIoNs on) ArbitraRy universe sets in thEIr sEmantics. Other LogIcs tHAT hAve A PrECisE sEManTICs, such as intuitiOnistic logIc, HAvE some additIOnaL sTructurEs, such aS topoLOgies, orDer relatiOns, accessIbIlitY RElaTions, etc. on The univeRse sets anD Thus tHOsE logiCs pArtake Of The SpeciFic natURe oF thosE additIoNal strUcturEs. 5 Same mathEmatical moves in the logic Of partItionS, i.e., By defininG a nORmaLized counTing Measure on tHe pArtItionS on A FinitE uniVErSe sET (wherE parTItions are REpResENTeD as the complEMENt oF the aSsoCIated eQuivAlence relations), onE Obtains a notion Of loGICal EntROpy ([11] aNd [13]) That adds to the iNteRpRETation of qM Mathematics.2 since our ToPIc is tO betteR underStand obJECtIVe indeFiniTenEss, and thuS to BeTTer
 subsets and partitions ar e thus ass ociat edwit htwoafor ementioned met a phys ical types of reality:  1. t he comm o n- sense  notion of o bje ct iv ely  d e fi niterea lity as sumed in c las si cal physics, an d 2. the n oti on of object ive ly ind ef ini t e rea lit y sug gested by qua ntum mech an i cs. Su b sets an d qu otie nt sets (or parti t io n s) are mathema ticall yd ua l con cep ts in there verse - thearro w ss e n seo f category-th eoretic dua l ity , e.g. ,a s u bset i s the  d i rec t image ofa se t monomor phismw hile as et part itionisthe  inv e rs eima ge ofa nepi m orp hism. Th is  d ualit y is f a m ilia r i n ab strac t algebra inthe  int e rpl ay of  subo bjec ts  (e.g ., sub group s,  subrings, etc. ) an d quotien t o bj ect s.  Will i am Law ver e c alls th e gener a l c at e g o ry -theoretic notionof a  s ubobject  a par t ,an d  then he  n ote s: " T h e dua l no t io n (obtai ned by re ve rsing t he  arrow s)  of  'p art'i s  the  notio n of par titio n ."[28, p. 85]T he logic appr o pr i a te forthe  usual noti on o f  ful ly d e fi nit e  real ity d es c ri b ed by subsets is th eordina ry Bo olean logic o f subsets[ 3 ]  (usuall y mi s -s p ecified as the  spec ial case o f  "propos ition al" logi c). We ha v e  seen th atthe  ot her v is ion of object i v elyin definit e r ealitysug ges ted  by  Q M is math ematical ly  d es cr ibe d byt he equiv al ent  c onc eptso f quot ientsets ,pa r tit ions, o r  e q u ival en ce  rel ati on s. Th e Bo o lea n logic  of subse tsh as a neq ually f undamental lo gi c of the d ua l q uotien t sets, eq uivalence relations, or partiti ons  ([12 ] an d [16]).1  Th e twolog i cs ofthe du al co nc ept s are a s s oc iat ed  with thet w o v ision sof r eality.  Boole developed n o t o nly the logic  of  sub s e ts , h e  d e fin ed a n o r malized countin g measureon su bsets of a fin it e unive rse set  to y i eld the  notion o f logical  f init e pro bability t heory. B y makingt he 1T h eBoole anlogicof  su bsets  and t h e l ogicof par ti tionsare e qu ally fun damental in that they a re the  twolog ics thattak e  (s ubsets of  and  partition s o n)arbit rar y  univ erse se tsi n the ir s e mantics.O th erl o gi cs that hav e a  pr ecise  se m antics , su ch as intuitionis t ic logic, have  som e add iti o nalst ructures, such  as  t o p ologies,  o rder relati ons, acc es s ibili ty rel ations , etc.o n  t h e univ erse  se ts and th usth o se logi cs  p a rtakeof t he  speci fic na t ureo f  those additiona l str u c tures .  5samema themati c al m oves in th e logic ofpartit ions , i.e ., by d ef ininga n or malized co u nting mea sureon thepa rtit ion s on a  fin i t e uni vers eset  (where p a r ti t io ns are  rep resen te d as  the comp l ement of  th e  associ at ede q uivale n ce r elations),  on e obt a i ns a notio n  ofl og i cal e ntropy  ([11]  and [1 3 ])th at adds  to t he interp retationo f Q Mmath ematics. 2Sinc eour  t opic is to bett er under stand ob j e ctiveind e fini t eness,  a nd thus to b etter 
 subsets_and partitions_are thus associated with_two aforementioned_metaphysical_types of_reality:_1. the common-sense_notion of objectively_definite reality assumed in_classical physics, and_2._the notion of objectively indefinite reality suggested by quantum mechanics. Subsets and quotient sets_(or_partitions) are_mathematically_dual_concepts in the reverse-thearrows sense_of category-theoretic duality, e.g., a_subset is_the direct image of a set monomorphism while_a_set partition is_the inverse image of an epimorphism. This duality is_familiar in abstract algebra in the_interplay of subobjects_(e.g.,_subgroups,_subrings, etc.) and quotient_objects. William Lawvere calls the general_category-theoretic notion of a subobject a_part, and then he notes: "The dual_notion (obtained by reversing the arrows)_of 'part'is the notion of_partition."[28, p._85] The logic appropriate for_the usual notion_of fully_definite reality described_by subsets is the ordinary Boolean_logic of subsets_[3] (usually mis-specified as the special_case_of "propositional" logic)._We_have_seen that_the other vision_of_objectively indefinite_reality_suggested by QM is mathematically described_by_the equivalent concepts of quotient sets, partitions,_or equivalence relations. The_Boolean_logic of subsets has_an equally fundamental logic of_the dual quotient sets, equivalence relations,_or partitions_([12] and_[16]).1 The two logics of the dual concepts are associated with_the two visions of reality. Boole_developed not only the_logic of_subsets,_he defined a_normalized_counting measure_on subsets of a finite universe set_to yield_the notion of logical finite probability_theory. By making the_1The_Boolean logic of subsets and the_logic of partitions are equally fundamental_in that they are the_two_logics_that take (subsets of and_partitions on) arbitrary universe sets in_their semantics. Other_logics that have a precise semantics, such_as_intuitionistic logic, have some additional structures,_such_as topologies, order relations, accessibility relations,_etc._on_the universe sets and thus_those logics partake of the specific_nature of those additional structures. 5 same mathematical moves_in the logic_of partitions, i.e., by defining_a_normalized_counting measure on the partitions on a finite universe set_(where partitions_are represented as_the complement of the associated equivalence relations), one obtains a_notion of logical entropy ([11] and [13])_that adds to the interpretation of QM mathematics.2 Since our topic_is to better understand objective indefiniteness, and thus_to better