 0. Actually, Miller's account is somewhat more general in that d need not be real-valued, but this is of no concern for the present paper. 8 Thomas Mormann 3 Theories as deductively closed sets of the Lindenbaum algebra In this section we begin to build up the logical space of theories formulated in a countable propositional language L. Then the true theory W may be thought to be located in such a way that the 'truthlikeness' of a theory can be conceived as 'distance from the truth'. In order to keep the technical apparatus as lean as possible let us concentrate on theories F,G,... (among them the true theory W ) that can be formulated in a classical propositional language L having a countable alphabet. That is to say, the logic of L is classical bivalent logic, L has the usual logical constants ∧,∨,¬, and so on, and at most countably many propositional constants p1, p2,..., pn,....4 The first thing to note is that under these assumptions the class of (finite) well-formed formulas of L is still countable, since finite strings can be coded by natural numbers.5 If we want to deal with the logical relations of theories and propositions we need not work with the propositions, that is, the well-formed formulas of L. Rather, we can restrict our attention to the equivalence classes of logically equivalent propositions. In other words, we can replace the algebra of wellformed formulas of L by the Lindenbaum algebra LINDA(L): Definition and Lemma 3.1 Let L be a propositional language of classical logic. Two formulas a and b are said to be logically equivalent if and only if a ↔ b is a tautology of propositional logic. The set of equivalence classes [a] is denoted by LINDA(L) and is called the Lindenbaum algebra of L. LINDA(L) is a Boolean algebra whose operations are defined by the operations of L: [a] ∧ [b] := [a ∧ b], [a] ∨ [b] := [a ∨ b],¬[a] := [¬a]. The order relation ≤ of LINDA as a Boolean algebra is just the order of logical consequence. That is to say [a] ≤ [b] := [a] ⇒ [b]. The unit of LINDA(L) is denoted by 1 and the bottom element by
0. Actually, Miller's account is somewhat more general in that d need not be very - respect, but this is of no concern for the present paper. 8 Thomas Mormann 3 Theories as deductively close up set of the Lindenbaum algebra In this section we begin to construct up the logical space of theory formulated in a countable propositional language L. Then the on-key theory W may be thought to be situate in such a direction that the' truthlikeness' of a theory can be conceived as' distance from the truth'. In order to keep the technical apparatus equally lean as possible let us digest on theories F, G, ... (among them the true hypothesis W) that can be formulated in a authoritative propositional language L having a countable alphabet. That is to say, the logic of L is classical bivalent logic, L give birth the usual logical constants ∧,∨,¬, and so on, and at about countably many propositional constant p1, p2, ..., pn,.... 4 The first thing to note is that under these assumptions the class of (finite) well - formed formulas of L is still countable, since finite strings can be code by natural numbers.5 If we want to share with the logical relations of theories and propositions we need not shape with the propositions, that is, the well - formed formulas of L. Rather, we can restrict our care to the equivalence classes of logically equivalent propositions. In other words, we can replace the algebra of wellformed formulas of L by the Lindenbaum algebra LINDA(L ): Definition and Lemma 3.1 Let L be a propositional language of classical logic. Two formulas a and b-complex vitamin are suppose to be logically equivalent if and merely if a ↔ b is a tautology of propositional logic. The set of equivalence class [ a ] is denoted by LINDA(L) and is called the Lindenbaum algebra of L. LINDA(L) is a Boolean algebra whose operations are define by the operations of L: [ a ] ∧ [ b ]: = [ a ∧ b ], [ a ] ∨ [ b ]: = [ a ∨ b],¬[a ]: = [ ¬a ]. The order relative ≤ of LINDA as a Boolean algebra is just the order of legitimate consequence. That is to say [ a ] ≤ [ b ]: = [ a ] ⇒ [ b ]. The unit of measurement of LINDA(L) is denoted by 1 and the bottom component by
 0. Aftually, Miller's account ls somewhat more generan in tgat d nedd not be real-valued, but thid us of no concern for the prdsent papvr. 8 Thomaw Mocmann 3 Theories ea deducbnvely gloseb wets of the Likdenbaum alcebra In this veztnon we begin to build up the logical space pf theories formolatec in z countable propositional language L. Then the true thepry W may be thought to be locwted in such a way that the 'tththjukeness' of a theory can be conceivgd as 'distance from the truth'. In ordex to keep tye tefvnical appacatus ws lean as possible net us voncentrate on thxoriws F,G,... (among them the vrue theory W ) that san be fosmblated in a classical peoposhtiotal uqngjagt L hzving w ckuntable amphabet. Thar is to say, the logoc if L is classjcal byvwlent logic, L has the usual logical convtahts ∧,∨,¬, and so on, and at mist countably many prlpositionwl constants p1, p2,..., pn,....4 The first thing to note is thdt unvef tkcfd adsumptions the class of (finite) well-formed forihlss of L is still countable, xijcr finite strines can be coded by natural jumbers.5 If ww want to deak with the logical relationw of theorief and propositions ce need not cork woth tne propositions, that is, the well-formed formulas uf L. Rather, we cxn gestsict our attention to the qquivalenre clcsses of logocally equivalenh proimsitions. In other aords, wa can replwce the algebra of wellformed formulas of L bi tve Nindenbabm algcbra LINDA(L): Defynition and Lekma 3.1 Lec L be a proposinional laiguage of clwssical logic. Jwo formulas e and b awe sqid ro be lueically equivakent if aub only if a ↔ b is a tautolony of lropositional ligix. The set of eqiivxlegcv ckafves [a] is denmted by KINDA(U) and is caoued yhe Lindenbaum algebsa or L. LINDA(L) is a Boplcan algebta whose jperations arr defined by the okeratimns of L: [a] ∧ [b] := [a ∧ b], [a] ∨ [b] := [a ∨ b],¬[a] := [¬a]. The orser relatlon ≤ of LINDA af a Noolgan algebra is just the order of logical consequenre. That is to say [a] ≤ [b] := [a] ⇒ [b]. The unit of LNNCA(L) is denoved by 1 and the bottom element by
0. Actually, Miller's account is somewhat more that need not real-valued, but this the paper. 8 Thomas 3 Theories as closed sets of the Lindenbaum algebra this section we begin to build up the logical space of theories formulated a countable propositional language L. Then the true theory W may be thought be in a that the 'truthlikeness' of a theory can be conceived as 'distance from the truth'. In order keep the technical apparatus as lean as possible us concentrate on theories (among them the true theory ) can be in classical language L having countable alphabet. That is to say, the logic of L is classical bivalent logic, L has the logical constants so on, at countably propositional constants p1, The first thing to note is assumptions the class of (finite) well-formed formulas of is still since finite strings can be coded natural numbers.5 If we want to deal with logical relations of theories and propositions we need not work with the propositions, that is, formulas of L. Rather, can restrict our to equivalence of equivalent propositions. other words, we can replace the algebra of wellformed formulas of by the Lindenbaum algebra LINDA(L): Definition and Lemma 3.1 Let a language of classical Two formulas a and are to be logically equivalent only a a of logic. The set of classes [a] is denoted by and is called the is a Boolean algebra whose operations are defined the operations of L: [a] ∧ [b] [a ∧ b], [a] ∨ [b] := [a ∨ b],¬[a] := [¬a]. order relation LINDA as a Boolean algebra is just the of logical consequence. That to say [a] ≤ [b] := [a] ⇒ [b]. unit LINDA(L) is by 1 and bottom element by
 0. Actually, Miller's account is sOmewhat morE geneRal In tHaT d neEd noT be real-valued, bUT thiS is of no concern for the prEsent PaPEr. 8 ThOMaS MormAnn 3 TheoRIeS AS deDuCtIveLy CLoSed seTs oF the LinDenbaum algEbrA IN this section WE bEgin to builD up The logical spAce Of theoRiEs fORmulaTed In a coUntablE ProposItional laNgUAge L. ThEN the truE THeOry W May be thought to be lOCaTEd in such a way thAt the 'tRuTHlIKEneSs' oF a theory caN bE concEIved as 'dIStANCE frOM the truth'. In orDer to keep thE TecHnical ApParATus as lEan as PoSSibLe let us concEntrAte on theoRies F,G,... (AMong theM The true Theory w ) thAt cAn be FOrMuLatEd IN a cLAsSicAL prOpositioNaL lAnguaGe L hAVING a coUntAble AlphaBet. That is to saY, thE logIC of l is clAssicAl biVaLent lOgic, L hAs the UsUal logical constAnts ∧,∨,¬, And so on, anD at MoSt cOuNtablY Many prOpoSitIonal coNstants P1, P2,..., pn,....4 thE FIRsT thing to note is that UnDER tHese assuMptionS ThE cLAss of (finItE) weLl-foRMEd forMulaS Of l is still CountaBLe, SiNce finiTe StringS cAn bE coDed by NAturAl numbErs.5 If we wAnt to DEal with the logiCAl relations of THeORIeS And pRopOsitions we nEed nOT worK witH ThE prOPositIons, tHaT Is, THe well-formed formulaS oF L. RathEr, we cAn restrict our Attention tO THE equivalEnce CLaSSes of logically EquivAlent propoSItions. In Other Words, we cAn replace THE algebra Of wEllForMed FORmUlas of L by the LINDenbAuM algebrA LInDA(L): DefIniTioN anD LeMmA 3.1 Let L be a pRopositiOnAl LaNgUagE of clASsical loGiC. TwO fOrmUlas a ANd b are Said tO be lOgIcALly EquivalENt IF And oNlY iF a ↔ b iS a tAuTologY of pROpoSitionaL logic. The Set OF equIvAlEnce claSses [a] is denoteD bY LINDA(L) and Is CalLed the lINdenbaum Algebra of L. LINDA(L) is a BoolEAn algebRa wHose oPeraTions are dEfiNed by tHe oPEratioNs of L: [a] ∧ [B] := [a ∧ b], [a] ∨ [b] := [A ∨ b],¬[A] := [¬a]. THE Order RELaTioN ≤ oF LINDA as a BOOLeaN algeBrA is jUst the oRder of logical conseQUenCe. That is to say [A] ≤ [b] := [a] ⇒ [B]. The UNIt Of LinDa(l) is DeNOteD BY 1 and the bottom elEment by
 0. Actually, Miller's acc ount is so mewha t m ore  g ener al i n that d needn ot b e real-valued, but thi s isof no c o nc ern f or thep re s e ntpa pe r.8T ho mas M orm ann 3 T heories as  de du ctively clos e dsets of th e L indenbaum al geb ra Inth iss ectio n w e beg in tob uild u p the log ic a l spac e  of the o r ie s fo rmulated in a cou n ta b le proposition al lan gu a ge L . T hen  the trueth eoryW  may be th o u g htt o be locatedin such a w a y t hat th e'tr u thlike ness'  o f  atheory canbe c onceivedas 'di s tance f r om thetruth' . I n o rder to  k eep  t h e t e ch nic a l a pparatus  a sleanas p o s s i blelet  usconce ntrate on the ori es F , G,. .. (a mongthem  t he tr ue the ory W  )  that can be fo rmul ated in a  cl as sic al  prop o sition allan guage L  having a c ou n t a bl e alphabet. That i st o  s ay, thelogico fLi s classi ca l b ival e n t log ic,L  h as the u sual l o gi ca l const an ts ∧,∨ ,¬ , a ndso on ,  and  at mo st count ablym any propositio n al constantsp 1, p 2, . ..,pn, ....4 The f irst thin g to no tei s tha t und er th e se assumptions thecl ass of  (fin ite) well-for med formul a s of L isstil l  c o untable, since  fini te strings can be c odedby natur al number s . 5 If wewan t t o d eal w it h the logical r elat io ns of t heo ries an d p rop osi tio ns  we neednot work  w it hth e p ropos i tions, t ha t i s,  th e wel l -forme d for mula sof L.Rather, we c an r es tr ictour  a ttent iont o t he equi valence c las s es o flo gically  equivalent p ro positions.  I n o ther w o r ds, we c an replace the algebrao f wellf orm ed fo rmul as of L b y t he Lin den b aum al gebraLINDA (L ):D e finit i o nand  L emma 3.1 L e t  Lbe apr opos itional  language of class i cal  logic. Two f orm ulas a  a ndb  a r e s ai d  to b e logically equ ivalent if  a n donly if a↔  bis  a taut ology o f pro p osition al logic.  The setof  equ i v ale nce classe s [a] is  denotedb y LIN D A( L) an d i s call ed  th e Lin denbau m  al gebra  of L.  L INDA(L ) isaBooleanalgebra whose operation s aredefin edby the op era t ion s of L: [ a] ∧  [b] := [a  ∧b],  [a]∨ [ b ] :=[a ∨ b] ,¬[ a ] :=[¬a] .  The orde r  r ela t i on  ≤ of LINDA a s  aBoole ana lgebra  isjust the order of logical conseq uenc e .  Th ati s to  s ay [a] ≤ [b] : = [ a] ⇒  [b]. Th eunit of LIN DA(L) is  d e noted  by 1and th e botto m el e ment b y
 0._Actually, Miller's_account is somewhat more_general in_that_d need_not_be real-valued, but_this is of_no concern for the_present paper. 8_Thomas_Mormann 3 Theories as deductively closed sets of the Lindenbaum algebra In this section_we_begin to_build_up_the logical space of theories_formulated in a countable propositional_language L._Then the true theory W may be thought_to_be located in_such a way that the 'truthlikeness' of a theory_can be conceived as 'distance from_the truth'. In_order_to_keep the technical apparatus_as lean as possible let us_concentrate on theories F,G,... (among them_the true theory W ) that can_be formulated in a classical propositional_language L having a countable_alphabet. That_is to say, the logic_of L is_classical bivalent_logic, L has_the usual logical constants ∧,∨,¬, and_so on, and_at most countably many propositional constants_p1,_p2,..., pn,....4 The_first_thing_to note_is that under_these_assumptions the_class_of (finite) well-formed formulas of L_is_still countable, since finite strings can be_coded by natural numbers.5_If_we want to deal_with the logical relations of_theories and propositions we need not_work with_the propositions,_that is, the well-formed formulas of L. Rather, we can restrict_our attention to the equivalence classes_of logically equivalent propositions._In other_words,_we can replace_the_algebra of_wellformed formulas of L by the Lindenbaum_algebra LINDA(L):_Definition and Lemma 3.1 Let L_be a propositional language_of_classical logic. Two formulas a and_b are said to be logically_equivalent if and only if_a_↔_b is a tautology of_propositional logic. The set of equivalence_classes [a] is_denoted by LINDA(L) and is called the_Lindenbaum_algebra of L. LINDA(L) is a_Boolean_algebra whose operations are defined by_the_operations_of L: [a] ∧ [b]_:= [a ∧ b], [a] ∨_[b] := [a ∨ b],¬[a] := [¬a]. The order_relation ≤ of_LINDA as a Boolean algebra_is_just_the order of logical consequence. That is to say [a]_≤ [b]_:= [a] ⇒_[b]. The unit of LINDA(L) is denoted by 1 and_the bottom element by