 by b. (iv)2 The theory F is a complete theory if & only if for all a ∈ LINDA(L) either a ∈ F or ¬a ∈ F. A complete theory is also called an ultrafilter.6 (iv)3 The theory F is consistent if & only if F 6= LINDA(L). The Boolean algebra LINDA(L) is called the trivial filter or the trivial theory on L. At first glance, it is not at all obvious that complete theories exist. Indeed, the existence of sufficiently many ultrafilters, that is, complete theories, presupposes the validity of the axiom of choice or some similar principle (see Davey & Priestley 1990, Chapter 9). The axiom of choice is assumed throughout the rest of this paper without further mention. Then the true theory W in the sense of Popper may be conceptualized as a distinguished ultrafilter, that is, as a distinguished complete theory W for which either a ∈ W or ¬a ∈ W for all a ∈ LINDA(L). There is a natural map r : LINDA(L) 7−→ T (L) defined by r(a) := {b | a ≤ b}. In order to render this map order-preserving one defines an order relation ≤ on T (L) by F ≤ G := G ⊆ F. Here, ⊆ is the standard settheoretical order defined on T (L) due to the fact that filters are subsets of LINDA(L). As is well known, T (L) endowed with this order ≤ is not just an ordered structure but is a complete co-Heyting algebra 〈T (L),≤〉 (Tarski 1935, Johnstone 1982).7 Since r : LINDA(L) 7−→ T (L) is order-preserving, one may consider the order relation ≤ on T (L) as a natural generalization of the logical order of propositions of LINDA(L). For 〈T (L),≤〉 one can define a theory of truthlikeness, or, more generally, a qualitative (order-theoretical) theory of theory comparison that is essen6Obviously, the notion of a filter can be defined for general lattices. For Boolean algebras the notions of ultrafilter and prime filter coincide. This does not hold for more general lattices: if LINDA(L) is not Boolean, ultrafilters are prime
by b. (iv)2 The theory F is a complete theory if & only if for all a ∈ LINDA(L) either a ∈ F or ¬a ∈ F. A accomplished hypothesis is also called an ultrafilter.6 (iv)3 The hypothesis F is consistent if & entirely if F 6= LINDA(L). The Boolean algebra LINDA(L) is called the trivial filter or the fiddling theory on L. At first glance, it is not at wholly obvious that accomplished theories exist. Indeed, the universe of sufficiently many ultrafilters, that is, complete theories, presuppose the cogency of the axiom of choice or some similar principle (see Davey & Priestley 1990, Chapter 9). The maxim of choice is assumed throughout the rest of this newspaper without further mention. Then the true hypothesis W in the sense of Popper may be gestate as a distinguished ultrafilter, that is, as a distinguished complete theory W for which either a ∈ W or ¬a ∈ W for all a ∈ LINDA(L). There is a natural map r: LINDA(L) 7−→ deoxythymidine monophosphate (L) defined by r(a): = { b | a ≤ b }. In order to render this function ordering - preserving one defines an order relation ≤ on T (L) by F ≤ G: = G ⊆ F. Here, ⊆ is the standard settheoretical order defined on T (L) due to the fact that filter are subsets of LINDA(L). As is well known, T (L) endow with this decree ≤ is not just an ordered structure but is a accomplished co - Heyting algebra 〈 T (L),≤ 〉 (Tarski 1935, Johnstone 1982).7 Since r: LINDA(L) 7−→ T (L) is order - preserving, one may view the order relation ≤ on T (L) as a natural generalization of the coherent order of propositions of LINDA(L). For 〈 T (L),≤ 〉 one can define a theory of truthlikeness, or, more by and large, a qualitative (order - theoretical) hypothesis of hypothesis comparison that is essen6Obviously, the impression of a filter can be defined for general lattices. For Boolean algebras the notions of ultrafilter and prime filter coincide. This does not hold for more general lattice: if LINDA(L) is not Boolean, ultrafilters are prime
 by b. (iv)2 The theory F is a gomplete theory nd & onlb if fod all a ∈ LINDA(L) either a ∈ F or ¬a ∈ F. E conplett theory is also cauled an uptrafiltwr.6 (it)3 The theory F is consisbznt ir & onnb if F 6= LINDA(L). Jhe Boolean dlgebra LINDA(L) ir ealled the trivial filter or the tridial throgy on L. At firft gkwnce, pt is not at all obvious that ckmplete theories exixt. Indeed, the existence of sufviciently many ultgafilters, tyat yw, complete tfeories, prtsbpposes the validity of the axiom of choicd or xome similqr prltciple (see Vavey & Priestley 1990, Gnapter 9). The aciom of choice is aswumed throughout the cest of this paper wythout fustker mention. Then the rrye thgory F in rhe sehsx or Poppfr jay be condeptualized as a distinguished ujntafilter, thaf is, af w distinguished complete theory W for wvicg either a ∈ W or ¬a ∈ W fir all a ∈ LINDA(L). Therg is a nattral map r : LINDA(L) 7−→ T (L) defined by r(a) := {b | a ≤ b}. In ordec go xckder rhls map order-preserving one defines an order rqmauiok ≤ on T (L) by F ≤ N := G ⊆ F. Here, ⊆ is tne sywndard settheureticcm krder defined on T (L) due jo the fact thau filyers are subsets of LINDA(L). Qs is well kuowb, T (L) endowed with this order ≤ is not kust an ordered structuxe but is a complfte co-Heyfkng algebra 〈T (L),≤〉 (Garxkh 1935, Johnsuune 1982).7 Since r : LINQA(L) 7−→ T (L) ms orber-presefvinb, one iay considfr thc order relation ≤ oj T (L) av a naturap generalization of the logical order of proppshtimns of LNNDA(L). Nor 〈T (L),≤〉 one can define a theoty of truchlikevess, or, moge generanly, a qualieative (order-tvgoretical) themry of treort conparisov that is essen6Pbviously, the notiob of a filter can ne dewjned for generao lqttices. For Bookeav ajgvbres thq notions of gltrxfiuyer avd primt flltdr cpincide. This does nod homd for more generak jattices: if LINDW(L) is not Bookean, ultrafilters wre pcime
by b. (iv)2 The theory F is theory & only for all a F ¬a ∈ F. complete theory is called an ultrafilter.6 (iv)3 The theory is consistent if & only if F 6= LINDA(L). The Boolean algebra LINDA(L) called the trivial filter or the trivial theory on L. At first glance, is at obvious complete theories exist. Indeed, the existence of sufficiently many ultrafilters, that is, complete theories, presupposes the of the axiom of choice or some similar (see Davey & Priestley Chapter 9). The axiom of is throughout the of paper further mention. Then true theory W in the sense of Popper may be conceptualized as a distinguished ultrafilter, that is, a distinguished W for either ∈ or ¬a ∈ all a ∈ LINDA(L). There is r : LINDA(L) 7−→ T (L) defined by := {b a ≤ b}. In order to this map order-preserving one defines an order relation on T (L) by F ≤ G := G ⊆ F. Here, ⊆ is the order defined on T due to the that are of As is known, T (L) endowed with this order ≤ is not just ordered structure but is a complete co-Heyting algebra 〈T (L),≤〉 Johnstone Since r : 7−→ T (L) is one consider the order relation T as of logical of propositions of LINDA(L). 〈T (L),≤〉 one can define theory of truthlikeness, or, theory of theory comparison that is essen6Obviously, the of a filter can be defined for lattices. For Boolean algebras the notions of ultrafilter and prime filter coincide. does not more general lattices: if LINDA(L) is not Boolean, are prime
 by b. (iv)2 The theory F is a completE theory if & oNly if For All A ∈ LiNDA(l) eitHer a ∈ F or ¬a ∈ F. A compLEte tHeory is also called an ultRafilTeR.6 (Iv)3 ThE ThEory F Is consiSTeNT If & oNlY iF F 6= LiNda(L). the BoOleAn algebRa LINDA(L) is CalLeD the trivial fILtEr or the triViaL theory on L. At FirSt glanCe, It iS Not at All ObvioUs that COmpletE theories ExISt. IndeED, the exiSTEnCe of Sufficiently many uLTrAFilters, that is, cOmpletE tHEoRIEs, pResUpposes the VaLiditY Of the axIOm OF CHoiCE or some similaR principle (sEE DaVey & PriEsTleY 1990, chapteR 9). The aXiOM of Choice is assUmed ThroughouT the reST of this PAper witHout fuRthEr mEntiON. THeN thE tRUe tHEoRy W IN thE sense of poPpEr may Be coNCEPTualIzeD as a DistiNguished ultraFilTer, tHAt iS, as a dIstinGuisHeD compLete thEory W FoR which either a ∈ W oR ¬a ∈ W fOr all a ∈ LINdA(L). thEre Is A natuRAl map r : lINdA(L) 7−→ t (L) definEd by r(a) := {b | A ≤ B}. In OrDER To Render this map order-PrESErVing one dEfines AN oRdER relatioN ≤ oN T (L) By F ≤ G := g ⊆ f. here, ⊆ iS the STaNdard setTheoreTIcAl Order deFiNed on T (l) dUe tO thE fact THat fIlters Are subseTs of LinDA(L). As is well knOWn, T (L) endowed wiTH tHIS oRDer ≤ iS noT just an ordeRed sTRuctUre bUT iS a cOMpletE co-HeYtINg ALgebra 〈T (L),≤〉 (Tarski 1935, JohnsToNe 1982).7 SincE r : LINdA(L) 7−→ T (L) is order-pReserving, oNE MAy considEr thE OrDEr relation ≤ on T (L) As a naTural generALization Of the Logical oRder of proPOSitions oF LInDA(l). FoR 〈T (L),≤〉 ONE cAn define a theoRY Of trUtHlikeneSs, oR, more geNerAllY, a qUalItAtive (ordeR-theoretIcAl) ThEoRy oF theoRY compariSoN thAt Is eSsen6OBViouslY, the nOtioN oF a FIltEr can be DEfINEd foR gEnEral LatTiCes. FoR BooLEan AlgebraS the notioNs oF UltrAfIlTer and pRime filter coiNcIde. This doeS nOt hOld for MORe generaL lattices: if LINDA(L) is not BOOlean, ulTraFilteRs arE prime
 by b. (iv)2 The theory Fis a compl ete t heo ryif  & o nlyif for all a ∈ LIND A(L) either a ∈ F or ¬ a ∈ F .A  com p le te th eory is al s o  ca ll ed  an  u l tr afilt er. 6 (iv)3  The theor y F  i s consistent if  & only if  F6= LINDA(L).  Th e Bool ea n a l gebra  LI NDA(L ) is c a lled t he trivia lf iltero r the t r i vi al t heory on L. At fi r st glance, it isnot at  a l lo b vio usthat compl et e the o ries ex i st . I nde e d, the existe nce of suff i cie ntly m an y u l trafil ters,  t h atis, complet e th eories, p resupp o ses the validit y of t heaxi om o f  c ho ice  o r  so m esim i lar  princip le  ( see D avey & P ries tle y 19 90, C hapter 9). Th e a xiom ofchoic e isassu me d thr oughou t the  r est of this pap er w ithout fu rth er  me nt ion.T hen th e t rue  theory  W in t h e s en s e of  Popper may be con ce p t ua lized as  a dis t in gu i shed ult ra fil ter, t hat i s, a s  a  disting uished co mp lete th eo ry W f or  wh ich  eith e r a∈ W or  ¬a ∈ Wfor a l l a ∈ LINDA(L) .  There is a n a tu r a lm ap r  :LINDA(L) 7− → T( L) d efin e dbyr (a) : = {b|a  ≤ b}. In order to ren de r this  maporder-preserv ing one de f i n es an or derr el a tion ≤ on T (L ) byF ≤ G := G ⊆ F. Her e, ⊆is the s tandard s e t theoreti cal  or der  de f i ne d on T (L) du e to t he  fact t hat  filter s a resub set sof LINDA( L). As i swe ll  k now n, T( L) endow ed  wi th  th is or d er ≤ i s not  jus tan ord ered st r uc t u re b ut  i s acom pl ete c o-He y tin g algeb ra 〈T (L) ,≤〉 (Tar sk i1935, J ohnstone 1982 ). 7 Since r:LIN DA(L)7 − → T (L)is order-preserving, on e  may co nsi der t he o rder rela tio n ≤ on  T( L) asa natu ral g en era l i zatio n of  th elogical or d e r o f pro po siti ons ofLINDA(L). For 〈T ( L ),≤ 〉 one can def ine  a t h e or y o f  t r uth li k ene s s , or, more gene rally, a q ua l it ative (ord e r-t he oretica l) theo ry of theorycompariso n that is  e ssen 6 O bvi ously, the  notionof a filt e r can be  defi ned  for g en era l lat tices. For  Bool ean al ge bras t he no ti ons of u ltrafilter and prime fi lter c oinci de.  This doe s n o t h old for m oregeneral la tti ces : ifLIN D A(L)is n o tBoo l ean,ultr a filters a r epri m e 
 by_b. (iv)2_The theory F is_a complete_theory_if &_only_if for all_a ∈ LINDA(L)_either a ∈ F_or ¬a ∈_F._A complete theory is also called an ultrafilter.6 (iv)3 The theory F is consistent_if_& only_if_F_6= LINDA(L). The Boolean algebra_LINDA(L) is called the trivial_filter or_the trivial theory on L. At first glance,_it_is not at_all obvious that complete theories exist. Indeed, the existence_of sufficiently many ultrafilters, that is,_complete theories, presupposes_the_validity_of the axiom of_choice or some similar principle (see_Davey & Priestley 1990, Chapter 9)._The axiom of choice is assumed throughout_the rest of this paper without_further mention. Then the true_theory W_in the sense of Popper_may be conceptualized_as a_distinguished ultrafilter, that_is, as a distinguished complete theory_W for which_either a ∈ W or ¬a_∈_W for all_a_∈_LINDA(L). There_is a natural_map_r :_LINDA(L)_7−→ T (L) defined by r(a)_:=_{b | a ≤ b}. In order_to render this map_order-preserving_one defines an order_relation ≤ on T (L)_by F ≤ G := G_⊆ F._Here, ⊆_is the standard settheoretical order defined on T (L) due to_the fact that filters are subsets_of LINDA(L). As is_well known,_T_(L) endowed with_this_order ≤_is not just an ordered structure but_is a_complete co-Heyting algebra 〈T (L),≤〉 (Tarski_1935, Johnstone 1982).7 Since_r_: LINDA(L) 7−→ T (L) is_order-preserving, one may consider the order_relation ≤ on T (L)_as_a_natural generalization of the logical_order of propositions of LINDA(L). For_〈T (L),≤〉 one_can define a theory of truthlikeness, or,_more_generally, a qualitative (order-theoretical) theory of_theory_comparison that is essen6Obviously, the notion_of_a_filter can be defined for_general lattices. For Boolean algebras the_notions of ultrafilter and prime filter coincide. This does_not hold for_more general lattices: if LINDA(L)_is_not_Boolean, ultrafilters are prime