n. This metric is equivalent to the standard metric in the sense that it induces the same topology on Rn. Hence, it is natural to ask if each metrical pospace admits a radially convex metric equivalent to the original metric. For compact pospaces, this question is answered in the affirmative by the following theorem of Carruth (1968, p. 229): Theorem 2.7 (Urysohn-Carruth metrization theorem) A compact pospace X with a metric d has an equivalent metric d? that is radially convex. The aim of this paper is to show that the logical space T (L) of theories of a countable language L is indeed a compact metric pospace, and therefore may be assumed to have a radially convex metric. As far as I know, the notion of a radially convex metric does not appear in the literature dealing with matters of truth approximation, but related notions have been discussed by several authors. In (1984) Miller develops a pseudo-metrical3 theory of Boolean algebras B that is based on the following two axioms for a realvalued function d : B×B 7−→ R (ibidem, p. 99): d(x, y) + d(y, z) ≥ d(x, z)(0) d(x, y) ≥ d(zx, zy) + d(x + z, x + y)(1) The inequality (0) is just the familiar 'triangle inequality' for distance functions. The meaning of (1) is more intricate. (1) may be interpreted as the essential link between the lattice structure and the metrical structure. As is shown by Miller, (0) and (1) imply that d is indeed a radially convex pseudometric which satisfies some further special properties. On the other hand, it seems that one cannot show that radial convexity entails (1). The relation between Miller's account and the one developed in this paper may roughly described as follows: Miller assumes that the Boolean algebra B is endowed with a real-valued function d satisfying (0) and (1). Then he shows that this function is a radially convex pseudo-metric. In this paper it is shown that any countable Boolean algebra can be endowed with a radially convex metric. Hence, we show that for countable Boolean algebras Miller's assumption can be partially satisfied. 3In contrast to a metrical distance function, for a pseudo-metric it may happen that for two distinct elements x and y one has d(x, y) =
n. This metric is equivalent to the standard metric in the sense that it induces the like regional anatomy on Rn. therefore, it is natural to ask if each metrical pospace admit a radially convex metric equivalent to the original metric. For compendious pospaces, this question is answered in the affirmative by the come theorem of Carruth (1968, p. 229 ): Theorem 2.7 (Urysohn - Carruth metrization theorem) A compact pospace X with a metric five hundred has an equivalent metric d? that is radially convex. The purpose of this paper is to show that the logical space T (L) of theory of a countable language L is indeed a compact measured pospace, and therefore may be assumed to receive a radially convex measured. As far as I know, the notion of a radially convex metric does not appear in the literature dealing with topic of truth approximation, but related notions have been discussed by several writer. In (1984) Miller develops a pseudo - metrical3 theory of Boolean algebras B that is based on the following two axioms for a realvalued function d: B×B 7−→ R (ibidem, p. 99 ): d(x, y) + d(y, z) ≥ d(x, z)(0) d(x, y) ≥ d(zx, zy) + d(x + z, x + y)(1) The inequality (0) is just the familiar' triangle inequality' for distance functions. The meaning of (1) is more intricate. (1) may be interpret as the essential link between the wicket social organization and the metrical structure. As is shown by Miller, (0) and (1) imply that vitamin d is indeed a radially convex pseudometric which satisfies some further special properties. On the other bridge player, it seems that one cannot show that radial convexity entails (1). The relation between Miller's account and the one develop in this paper may roughly described as follows: Miller assume that the Boolean algebra B is endow with a real - valued function vitamin d satisfying (0) and (1). Then he shows that this function is a radially convex pseudo - measured. In this paper it is prove that any countable Boolean algebra can be endowed with a radially convex metric. therefore, we show that for countable Boolean algebras Miller's assumption can be partially satisfied. 3In contrast to a metrical distance affair, for a pseudo - measured it may happen that for two distinct elements x and y one has d(x, y) =
n. Tjis metric is equivalent to the standarb metrir in ths sense ghat it induces the same toplligy ob Rn. Hence, it is naturxl to ask if each metcical pospace admits a radiallg conree metric equivakent to tha original metsiz. Yor compact pospaces, this question if answetef in the affiriatine by fhe following theorem of Carruth (1968, l. 229): Theogem 2.7 (Urysohn-Carruyh metrization theorem) A clmpaft pospace X with w metric d yas wb equivalent metric d? uhct is radiamly convex. The aim of this papef is co show thaj fhf logical spece T (J) of theories of a cmuntablr language L ix iideee a compact metric povpace, and thereforg may be avsbmed to have a radialoy convgx medric. Qs war ax J know, thx notion of a radially convex metric does njn appear in tge litqrwture dealing with matters of truth appgoxijation, but related notiins have been discussgd by sevewal authors. In (1984) Miller develops a pseudo-metrical3 dheorb uf Yioleav apgebras B that is based on the following two wsipmx for a realvajued functipn d : B×B 7−→ R (ibidem, p. 99): d(x, y) + d(y, z) ≥ d(x, z)(0) d(x, y) ≥ f(zx, zy) + d(x + z, x + y)(1) Tht ineauality (0) is just the familiqr 'triangle pneqyality' for distancz functions. Che mesning of (1) is more intricate. (1) may be interprfted as tgd essential link benweet the laugice structure anq the metcical structjre. Ss is fhown by Mlller, (0) and (1) imply that d is iudeed a radiallj convex pseudometric which satmxfies some fortver special propcrties. On the oeher hand, it sgems that one zannot shos that cadial convevity entails (1). Jhe relation uetween Myllee's axcount xvd the one devrloped in this papee may roughly descvibed zs follows: Millzx qssumes that thr Buolqaj elgebwd B is endowad wkth s reau-valued fungtiun d satisfying (0) and (1). Than hs shows that this gukction is a radiajly convex psrudo-metric. In this papec it ir shoen jhat any countable Boolean algegra can bf ekdowed with a radlalli convex mecric. Hence, we show that for countable Boilean algebras Miloer's assumption can ne partially satiffied. 3In wontrast to a metricql distance functlon, for a pseudo-metric it maf hapoen that for two distinct elements x and y one has d(x, y) =
n. This metric is equivalent to the in sense that induces the same is to ask if metrical pospace admits radially convex metric equivalent to the metric. For compact pospaces, this question is answered in the affirmative by the theorem of Carruth (1968, p. 229): Theorem 2.7 (Urysohn-Carruth metrization theorem) A compact X a d an equivalent metric d? that is radially convex. The aim of this paper is to show the logical space T (L) of theories of countable language L is a compact metric pospace, and may assumed to a convex As far as know, the notion of a radially convex metric does not appear in the literature dealing with matters truth approximation, notions have discussed several In (1984) Miller pseudo-metrical3 theory of Boolean algebras B on the following two axioms for a realvalued d : 7−→ R (ibidem, p. 99): d(x, + d(y, z) ≥ d(x, z)(0) d(x, y) d(zx, zy) + d(x + z, x + y)(1) The inequality (0) is just the inequality' for distance functions. meaning of (1) more (1) be as the link between the lattice structure and the metrical structure. As is by Miller, (0) and (1) imply that d is indeed convex which satisfies some special properties. On the hand, seems that one cannot radial entails between account the one developed in paper may roughly described as Miller assumes that the with a real-valued function d satisfying (0) and Then he shows that this function is radially convex pseudo-metric. In this paper it is shown that any countable algebra can with a radially convex metric. Hence, we show for countable Boolean algebras assumption can be partially satisfied. 3In contrast to a distance for a it may happen for two distinct x and y d(x, y)
n. This metric is equivalent to The standarD metrIc iN thE sEnse That It induces the saME topOlogy on Rn. Hence, it is natuRal to AsK If eaCH mEtricAl pospaCE aDMIts A rAdIalLy COnVex meTriC equivaLent to the oRigInAl metric. For cOMpAct pospaceS, thIs question is AnsWered iN tHe aFFirmaTivE by thE folloWIng theOrem of CarRuTH (1968, p. 229): TheoREm 2.7 (UrysoHN-caRrutH metrization theorEM) A COmpact pospace X With a mEtRIc D HAs aN eqUivalent meTrIc d? thAT is radiALlY CONveX. the aim of this pAper is to shoW ThaT the loGiCal SPace T (L) Of theOrIEs oF a countable LangUage L is inDeed a cOMpact meTRic pospAce, and TheRefOre mAY bE aSsuMeD To hAVe A raDIalLy convex MeTrIc. As fAr as i KNOW, the NotIon oF a radIally convex meTriC doeS Not AppeaR in thE litErAture DealinG with MaTters of truth appRoxiMation, but RelAtEd nOtIons hAVe been DisCusSed by seVeral auTHorS. IN (1984) mILlEr develops a pseudo-mEtRICaL3 theory oF BooleAN aLgEBras B thaT iS baSed oN THe folLowiNG tWo axioms For a reALvAlUed funcTiOn d : B×B 7−→ R (IbIdeM, p. 99): d(X, y) + d(y, z) ≥ D(X, z)(0) d(x, Y) ≥ d(zx, zy) + D(x + z, x + y)(1) The InequALity (0) is just the fAMiliar 'trianglE InEQUaLIty' fOr dIstance funcTionS. the mEaniNG oF (1) is MOre inTricaTe. (1) MAy BE interpreted as the esSeNtial lInk beTween the lattiCe structurE AND the metrIcal STrUCture. As is shown By MilLer, (0) and (1) implY That d is iNdeed A radiallY convex psEUDometric WhiCh sAtiSfiES SoMe further specIAL proPeRties. On The Other haNd, iT seEms ThaT oNe cannot sHow that rAdIaL cOnVexIty enTAils (1). The rElAtiOn BetWeen MILler's aCcounT and ThE oNE deVeloped IN tHIS papEr MaY rouGhlY dEscriBed aS FolLows: MilLer assumeS thAT the boOlEan algeBra B is endowed WiTh a real-valUeD fuNction D SAtisfyinG (0) and (1). Then he shows that this FUnction Is a RadiaLly cOnvex pseuDo-mEtric. IN thIS paper It is shOwn thAt Any COUntabLE boOleAn Algebra can BE EndOwed wItH a raDially cOnvex metric. Hence, we SHow That for countaBle boolEAN aLgeBRaS milLeR'S asSUMption can be partIally satisFiED. 3IN contrast tO A meTrIcal disTance fuNctioN, For a pseUdo-metric It may happEn That FOR twO distinct eLements x And y one haS D(x, y) =
n. This metric is equivale nt to thestand ard  me tr ic i n th e sense that i t  ind uces the same topology  on R n. Henc e ,it is  natura l  t o ask  i feac hm et rical  po space a dmits a ra dia ll y convex met r ic  equivalen t t o the origin almetric .For compa ctpospa ces, t h is que stion isan s weredi n the a f f ir mati ve by the followi n gt heorem of Carr uth (1 96 8 ,p .  22 9):  Theorem 2 .7  (Ury s ohn-Car r ut h m etr i zation theore m) A compac t  po spaceXwit h  a met ric d  h a s a n equivalen t me tric d? t hat is radiall y  convex . Theaim  of  thi s  p ap eris tos ho w t h atthe logi ca lspace  T ( L ) o f th eor iesof acountable lan gua ge L isindee d a c ompa ct  metr ic pos pace,  a nd therefore ma y be  assumedtoha vearadia l ly con vex  me tric. A s far a s  Ikn o w ,  t he notion of a rad ia l l yconvex m etricd oe sn ot appea rinthel i terat ured ea ling wit h matt e rs  o f truth  a pproxi ma tio n,but r e late d noti ons have  been discussed by s e veral authors .  I n (1 9 84)Mil ler develop s ap seud o-me t ri cal 3  theo ry of  B o ol e an algebras B thatis  based  on t he followingtwo axioms f o r a real valu e df unction d : B× B 7−→  R (ibidem ,  p. 99):  d(x,  y) + d( y, z) ≥ d ( x , z)(0)d(x , y ) ≥  d( z x ,zy) + d(x + z , x +y) (1) The  in equalit y ( 0)isjus tthe famil iar 'tri an gl ein equ ality '  for dis ta nce  f unc tions .  The m eanin g of  ( 1) ismore in t ri c a te.(1 )maybein terpr eted asthe ess ential li nkb etwe en  t he latt ice structure  a nd the met ri cal  struc t u re. As i s shown by Miller, (0)a nd (1)imp ly th at d  is indee d a  radia lly convex  pseud ometr ic  wh i c h sat i s fi esso me further s pec ial p ro pert ies. On  the other hand, i t  se ems that onecan nots h ow  th a tr adi al con v e xity entails (1 ). The rel at i on  between M i lle r' s accou nt andthe o n e devel oped in t his paper  m ay r o u ghl y describe d as fol lows: Mil l er as s um es th atthe Bo ol ean  alge bra Bi s e ndowe d with  a  real- value dfunction  d satisfying (0) and ( 1). Th en he  sh ows thatthi s  fu nction is  a r adially co nve x p seudo -me t ric.In t h is  pa p er it  iss hown that an y c o u nt able Boolea n a lge bra c anb e endo wedwith a radially c o nvex metric. H ence , wesho w  tha tfor countableBoo le a n  algebra sMiller's as sumption  c a n bepartia lly sa tisfied . 3I n  contr asttoa metrica l d is t ance fu nc ti o n, for  a p se udo-me tric i t  may h appen that for t wo di s t incte lem entsxand y o n e ha s d(x, y)=
n. This_metric is_equivalent to the standard_metric in_the_sense that_it_induces the same_topology on Rn._Hence, it is natural_to ask if_each_metrical pospace admits a radially convex metric equivalent to the original metric. For compact_pospaces,_this question_is_answered_in the affirmative by the_following theorem of Carruth (1968,_p. 229):_Theorem 2.7 (Urysohn-Carruth metrization theorem) A compact pospace_X_with a metric_d has an equivalent metric d? that is radially_convex. The aim of this paper_is to show_that_the_logical space T (L)_of theories of a countable language_L is indeed a compact metric_pospace, and therefore may be assumed to_have a radially convex metric. As_far as I know, the_notion of_a radially convex metric does_not appear in_the literature_dealing with matters_of truth approximation, but related notions_have been discussed_by several authors. In (1984) Miller_develops_a pseudo-metrical3 theory_of_Boolean_algebras B_that is based_on_the following_two_axioms for a realvalued function d_:_B×B 7−→ R (ibidem, p. 99): d(x,_y) + d(y, z)_≥_d(x, z)(0) d(x, y)_≥ d(zx, zy) + d(x_+ z, x + y)(1) The_inequality (0)_is just_the familiar 'triangle inequality' for distance functions. The meaning of (1)_is more intricate. (1) may be_interpreted as the essential_link between_the_lattice structure and_the_metrical structure._As is shown by Miller, (0) and_(1) imply_that d is indeed a radially_convex pseudometric which satisfies_some_further special properties. On the other_hand, it seems that one cannot_show that radial convexity entails_(1)._The_relation between Miller's account and_the one developed in this paper_may roughly described_as follows: Miller assumes that the Boolean_algebra_B is endowed with a real-valued_function_d satisfying (0) and (1). Then_he_shows_that this function is a_radially convex pseudo-metric. In this paper_it is shown that any countable Boolean algebra can_be endowed with_a radially convex metric. Hence,_we_show_that for countable Boolean algebras Miller's assumption can be partially_satisfied. 3In_contrast to a_metrical distance function, for a pseudo-metric it may happen that_for two distinct elements x and y_one has d(x, y) =