 filters, but the reverse does not hold. 7These authors consider the set-theoretical order ⊆ on T (L) that is opposite to the 'logical' order ≤. Hence for them, the structure of T (L) is that of a Heyting algebra. 10 Thomas Mormann tially equivalent to the 'naive approach' of Kuipers (2002). This can be seen as follows: recall that the intersection ⋂ Fi of an arbitrary family of filters is again a filter. More precisely, it is the supremum of the Fi with respect to the order relation ≤. Hence, for any subset S ⊆ LINDA(L) there is a maximal filter F with F ≤ S, to wit, the set-theoretical intersection of all filters including S. Hence, for any filter F its quasi-complement F# is defined by F# := ⋂ {Fi | CF ⊆ Fi}, CF being the set-theoretical complement of F. Moreover, for two filters F and G there is a unique largest filter F ∨ G included in both, to wit, the intersection of all filters that include F and G. Hence one may define: Definition 3.3 Let U, V, and R be theories on L. Then U is at least as close to the target theory R as V is (U ≤R V ) if and only if (U ∩R#) ∪ (U# ∩R) ⊆ (V ∩R#) ∪ (V # ∩R) This definition corresponds to Kuipers's 'naive approach'. The only difference is that T (L) is not a Boolean algebra of sets but only a complete co-Heyting algebra. This means essentially that the quasi-complement V # of V satisfies only some weaker versions of the familiar laws of a Boolean complement: (U ∧ U#)# = 1 U## ≤ U U# = U###(3.4) Since Definition 3.3 depends only on the logical structure, one may call it an objective qualitative measure of similarity. Or, in the case the target theory R is the true theory W, the relation ≤W defines an objective logical measure of truthlikeness.8 The shortcomings of this 'naive account' are well-known and need not be rehearsed here (cp. Miller 1984, Kuipers 2000, Chapter 10). As an important flaw of Definition 3.3 I take the fact that
filters, but the reverse does not hold. 7These authors think the rig - theoretical order ⊆ on T (L) that is opposite to the' logical' orderliness ≤. Hence for them, the structure of T (L) is that of a Heyting algebra. 10 Thomas Mormann tially equivalent to the' primitive approach' of Kuipers (2002). This can be seen as come: recall that the intersection ⋂ Fi of an arbitrary syndicate of filters is again a filter. More precisely, it is the supremum of the Fi with respect to the club relation ≤. therefore, for any subset S ⊆ LINDA(L) there is a maximal filter F with F ≤ S, to brain, the fit - theoretical intersection of all filters including S. Hence, for any filter F its quasi - complement farad # is defined by F #: = ⋂ { Fi | CF ⊆ Fi }, CF being the set - theoretical complement of F. Moreover, for two filters F and G there be a unique largest filter F ∨ G included in both, to wit, the overlap of all filters that include F and G. Hence one may define: Definition 3.3 Let U, V, and radius be theories on L. Then U is at least as close to the target theory R as V is (U ≤R V) if and only if (uracil ∩R #) ∪ (U # ∩R) ⊆ (V ∩R #) ∪ (five # ∩R) This definition corresponds to Kuipers's' naive approach'. The only remainder is that T (L) is not a Boolean algebra of sets but only a complete co - Heyting algebra. This mean essentially that the quasi - complement V # of V satisfies only some weaker versions of the conversant laws of a Boolean complement: (U ∧ U #) # = 1 U # # ≤ U U # = U###(3.4) Since Definition 3.3 depends merely on the logical structure, one may call it an objective qualitative measure of similarity. Or, in the case the prey theory R is the true hypothesis W, the relation ≤W define an objective logical measure of truthlikeness.8 The defect of this' naive account' are well - known and need not be rehearsed here (cp. Miller 1984, Kuipers 2000, Chapter 10). As an important defect of Definition 3.3 I take the fact that
 fipters, but the reverse dots not hold. 7These authorv consjder the set-theoretical order ⊆ on T (P) rhat us opposite to the 'logkcal' ordeg ≤. Hence dor uhem, the structurx of T (L) is thzb of c Ieyting algebra. 10 Thomas Mmrmann tially aqjiralent to the 'naive approach' of Kuipqrs (2002). Thos can be seen af fokjows: gegall that the intersection ⋂ Fi or an aruitrary family pf filters is again a filtfr. Mlre precisely, it id the suprenum jd the Fi witf respect uo the order relation ≤. Hence, for any subset S ⊆ LNNDA(L) there iw a kaximal filver F rith F ≤ S, to wit, the set-thepretical interxecvion of all filters incluving S. Hence, for any filter F ics quasi-complement F# us defited ty F# := ⋂ {Fk | DF ⊆ Ri}, CF heiig the set-tgeoretical xomplement of F. Mortovqg, for two filfers F agd G there is a unique largest filter F ∨ G included in both, to wir, the intersection of all filtqrs that include F and G. Hence one may define: Defhnitikv 3.3 Oeb U, X, ajd R be theories on L. Then U is at least as cjkst tp the target tmeory R as V is (U ≤T G ) yf and only iw (U ∩R#) ∪ (U# ∩D) ⊆ (V ∩R#) ∪ (V # ∩R) This fefinityon cirrespondf to Kuipers's 'naive approach'. Thw only diffegencw is that T (L) is nlt a Booleau algenra og sets but only a complzte co-Geyting algfbra. This oeans essentiallh tmat the quasi-complement V # of V satisfmes ouly some wealer vewsions of hhe familiar laws of a Bloleau com[lement: (U ∧ U#)# = 1 U## ≤ U U# = U###(3.4) Since Definition 3.3 depends only ot tve logiccl strmcture, one may sall it an objgctive quclitatkve measurv of siminarity. Or, ig the case tha target theocy R is tre teue rheory D, the relation ≤E defines an objectuve logical measurc of jrhthlikeness.8 The shirtcomings of tnis 'nayvv arcoune' are well-knofn avd vred nut be rtkeavsea hete (cp. Miller 1984, Kuiperv 2000, Cgapter 10). As an impottwnt flaw of Defigition 3.3 I takr the fact that
filters, but the reverse does not hold. consider set-theoretical order on T (L) 'logical' ≤. Hence for the structure of (L) is that of a Heyting 10 Thomas Mormann tially equivalent to the 'naive approach' of Kuipers (2002). This be seen as follows: recall that the intersection ⋂ Fi of an arbitrary of is a More precisely, it is the supremum of the Fi with respect to the order relation ≤. for any subset S ⊆ LINDA(L) there is maximal filter F with ≤ S, to wit, the intersection all filters S. for filter F its F# is defined by F# := ⋂ {Fi | CF ⊆ Fi}, CF being the set-theoretical complement F. Moreover, filters F G is unique largest filter G included in both, to wit, all filters that include F and G. Hence may define: 3.3 Let U, V, and R theories on L. Then U is at least close to the target theory R as V is (U ≤R V ) if and (U ∩R#) ∪ (U# ⊆ (V ∩R#) (V ∩R) definition to Kuipers's approach'. The only difference is that T (L) is not a algebra of sets but only a complete co-Heyting algebra. This that quasi-complement V # V satisfies only some versions the familiar laws of complement: ∧ U## U = U###(3.4) Since Definition depends only on the logical one may call it similarity. Or, in the case the target theory is the true theory W, the relation defines an objective logical measure of truthlikeness.8 The shortcomings of this 'naive are well-known not be rehearsed here (cp. Miller 1984, Kuipers Chapter 10). As an flaw of Definition 3.3 I take the fact that
 filters, but the reverse does nOt hold. 7ThesE authOrs ConSiDer tHe seT-theoretical orDEr ⊆ on t (L) that is opposite to the 'lOgicaL' oRDer ≤. HENcE for tHem, the sTRuCTUre Of t (L) Is tHaT Of A HeytIng Algebra. 10 thomas MormAnn TiAlly equivaleNT tO the 'naive aPprOach' of KuiperS (2002). ThIs can bE sEen AS follOws: RecalL that tHE interSection ⋂ Fi Of AN arbitRAry famiLY Of FiltErs is again a filter. mOrE Precisely, it is tHe suprEmUM oF THe FI wiTh respect tO tHe ordER relatiON ≤. HENCE, foR Any subset S ⊆ LINdA(L) there is a MAxiMal filTeR F wITh F ≤ S, to Wit, thE sET-thEoretical inTersEction of aLl filtERs incluDIng S. HenCe, for aNy fIltEr F iTS qUaSi-cOmPLemENt f# is DEfiNed by F# := ⋂ {Fi | cF ⊆ fi}, cF beiNg thE SET-TheoRetIcal ComplEment of F. MoreoVer, For tWO fiLters f and G TherE iS a uniQue larGest fIlTer F ∨ G included in Both, To wit, the iNteRsEctIoN of alL FilterS thAt iNclude F And G. HenCE onE mAY DEfIne: Definition 3.3 Let U, V, AnD r Be Theories On L. TheN u iS aT Least as cLoSe tO the TARget tHeorY r aS V is (U ≤R V ) iF and onLY iF (U ∩r#) ∪ (U# ∩R) ⊆ (V ∩R#) ∪ (V # ∩R) thIs defiNiTioN coRrespONds tO KuipeRs's 'naive ApproACh'. The only diffeREnce is that T (L) iS NoT A boOLean AlgEbra of sets bUt onLY a coMpleTE cO-HeYTing aLgebrA. THIs MEans essentially that ThE quasi-ComplEment V # of V satiSfies only sOME Weaker veRsioNS oF The familiar lawS of a BOolean compLEment: (U ∧ U#)# = 1 U## ≤ u U# = U###(3.4) SiNce DefinItion 3.3 depeNDS only on tHe lOgiCal StrUCTuRe, one may call iT AN objEcTive quaLitAtive meAsuRe oF siMilArIty. Or, in thE case the TaRgEt ThEorY R is tHE true theOrY W, tHe RelAtion ≤w DefineS an obJectIvE lOGicAl measuRE oF TRuthLiKeNess.8 the ShOrtcoMingS Of tHis 'naivE account' aRe wELl-knOwN aNd need nOt be rehearsed HeRe (cp. Miller 1984, kuIpeRs 2000, ChapTER 10). As an impOrtant flaw of Definition 3.3 I TAke the fAct That
 filters, but the reversedoes not h old.7Th ese  a utho rs c onsider the se t -the oretical order ⊆ on T(L) t ha t  iso pp osite  to the 'l o g ica l'  o rde r≤ .Hence  fo r them,  the struc tur eof T (L) ist ha t of a Hey tin g algebra. 1 0 T homasMo rma n n tia lly  equi valent to the  'naive a pp r oach'o f Kuipe r s  ( 2002 ). This can be se e na s follows: rec all th at th e int ers ection ⋂ F iof an arbitra r yf a m ily of filters is  again a fi l ter . More  p rec i sely,it is  t h e s upremum oftheFi with r espect to theo rder re lation  ≤.  He nce, fo rany  s u bse t  S  ⊆L IND A(L) the re  i s a m axim a l f ilte r F  wit h F ≤  S, to wit, t heset- t heo retic al in ters ec tionof all  filt er s including S.Henc e, for an y f il ter  F  itsq uasi-c omp lem ent F#is defi n edby F #  : = ⋂ {Fi | CF ⊆ Fi} ,C F  b eing the  set-t h eo re t ical com pl eme nt o f F. Mo reov e r,  for two  filte r sFand G t he re isauni que  larg e st f ilterF ∨ G in clude d  in both, to w i t, the inters e ct i o no f al l f ilters that  inc l udeF an d  G . H e nce o ne ma yd ef i ne: Definition 3.3Le t U, V , and  R be theorie s on L. Th e n U is atleas t  a s  close to thetarge t theory R as V is(U ≤R  V ) ifand onlyi f  (U ∩R#)  ∪(U#  ∩R ) ⊆ ( V∩R#) ∪ (V # ∩ R )  Thi sdefinit ion  corres pon dstoKui pe rs's 'nai ve appro ac h' .Th e o nly d i fference  i s t ha t T  (L)i s nota Boo lean  a lg e bra  of set s  b u t  onl yacomp let eco-He ytin g  al gebra.This mean s e s sent ia ll y thatthe quasi-com pl ement V #of  Vsatisf i e s only s ome weaker versions oft he fami lia r law s of  a Boolea n c omplem ent :  (U ∧U#)# =  1 U# #≤ U U # = U # # #( 3.4 )Since Defi n i tio n 3.3  d epen ds only  on the logical st r uct ure, one maycal l it a nobj e ct i vequ a lit a t ive measure ofsimilarity .O r,  in the ca s e t he  target  theory  R is the tru e theoryW, the re la tion ≤ W d efines anobjectiv e logical measu r eof tr uth likene ss .8The s hortco m ing s ofthis ' na ive ac count 'are well -known and need not berehear sed h ere  (cp. Mil ler 198 4, Kuiper s 20 00, Chapte r 1 0).  As a n i m porta nt f l aw  of Defin itio n  3.3 I ta k ethe f ac t that
 filters,_but the_reverse does not hold._7These authors_consider_the set-theoretical_order_⊆ on T_(L) that is_opposite to the 'logical'_order ≤. Hence_for_them, the structure of T (L) is that of a Heyting algebra. 10 Thomas_Mormann_tially equivalent_to_the_'naive approach' of Kuipers (2002)._This can be seen as_follows: recall_that the intersection ⋂ Fi of an arbitrary_family_of filters is_again a filter. More precisely, it is the supremum_of the Fi with respect to_the order relation_≤._Hence,_for any subset S_⊆ LINDA(L) there is a maximal_filter F with F ≤ S,_to wit, the set-theoretical intersection of all_filters including S. Hence, for any_filter F its quasi-complement F#_is defined_by F# := ⋂ {Fi_| CF ⊆_Fi}, CF_being the set-theoretical_complement of F. Moreover, for two_filters F and_G there is a unique largest_filter_F ∨ G_included_in_both, to_wit, the intersection_of_all filters_that_include F and G. Hence one_may_define: Definition 3.3 Let U, V, and_R be theories on_L._Then U is at_least as close to the_target theory R as V is_(U ≤R_V )_if and only if (U ∩R#) ∪ (U# ∩R) ⊆ (V_∩R#) ∪ (V # ∩R) This_definition corresponds to Kuipers's_'naive approach'._The_only difference is_that_T (L)_is not a Boolean algebra of sets_but only_a complete co-Heyting algebra. This means_essentially that the quasi-complement_V_# of V satisfies only some_weaker versions of the familiar laws_of a Boolean complement: (U_∧_U#)#_= 1 U## ≤ U_U# = U###(3.4) Since Definition 3.3_depends only on_the logical structure, one may call it_an_objective qualitative measure of similarity. Or,_in_the case the target theory R_is_the_true theory W, the relation_≤W defines an objective logical measure_of truthlikeness.8 The shortcomings of this 'naive account' are_well-known and need_not be rehearsed here (cp._Miller_1984,_Kuipers 2000, Chapter 10). As an important flaw of Definition_3.3 I_take the fact_that