 of a. More precisely, t(R(a)) ∩ m(V (α)) and R(b) are open neighbourhoods of a and b such that for all x ∈ t(R(a)) ∩m(V (α)) and y ∈ t(R(b)) we have NOT (x ≤ y). The reason is simply that the elements y of R(b) do not contain the point α while the elements x of t(R(a)) ∩m(V (α)) are open neighbourhoods of α by definition. Hence, C(X) is a pospace. Thanks to Theorem 5.3, we may apply the Urysohn-Carruth metrization theorem (2.4) to the space of theories 〈C(ST (L)),≤〉 obtaining a radially convex metric d on C(ST (L)). Putting things together we finally have reached the main theorem of this paper: Theorem 5.4 Let L be a countable language. Then the logical space T (L) of theories on L can be conceived of as the Vietoris space C(ST (L)) of closed sets of ST (L).10 Then T (L) is a pospace that can be endowed with a radially convex metric. This metric can be used for truth approximation, or, more generally, for a metrical theory of theory comparison. It is, as it should be, in line with the underlying topological structure, and at least partially with the relevant logical structure, to wit the order structure ≤. For the time being, I do not know if it satisfies the further condition Miller (1984) set up for a good metric of truthlikeness In order to understand how this deficit may be overcome it is useful to dwell in some more detail upon Carruth's proof of his metrization theorem. It mimics exactly the proof of the classical theorem of Urysohn according to which a second countable compact Hausdorff space X is metrizable 10In a somewhat different manner, Brink, Pretorius, & Vermeulen (1992) use the order structure of the Vietoris space to define an order-theoretical notion of truth approximation. The main difference to the present paper is that they do not consider the possibility of endowing T(L) with a (radially convex) metrical structure. 14 Thomas Mormann since there is an embedding e : X 7−→ IN. Explicitly,
of a. More precisely, t(R(a) ) ∩ m(V (α) ) and R(b) are open neighbourhoods of a and b such that for all x ∈ t(R(a) ) ∩m(V (α) ) and y ∈ t(R(b) ) we have NOT (x ≤ y). The reason is plainly that the chemical element y of R(b) do not contain the point α while the elements x of t(R(a) ) ∩m(V (α) ) are assailable neighbourhoods of α by definition. Hence, C(X) is a pospace. Thanks to Theorem 5.3, we may give the Urysohn - Carruth metrization theorem (2.4) to the distance of theories 〈 C(ST (L)),≤ 〉 obtaining a radially convex metric d on C(ST (L) ). frame things together we finally have reached the main theorem of this paper: Theorem 5.4 permit L be a countable language. Then the logical quad T (L) of theories on L can be conceived of as the Vietoris outer space C(ST (L) ) of closed sets of ST (L).10 Then T (L) is a pospace that can be endowed with a radially convex metric. This metric can be use for truth approximation, or, more generally, for a metrical theory of hypothesis comparison. It is, as it should be, in channel with the underlying topological structure, and at least partially with the relevant logical structure, to wit the order structure ≤. For the time being, I do not know if it satisfies the further circumstance Miller (1984) sic up for a good system of measurement of truthlikeness In order to understand how this deficit may be overcome it is useful to harp in some more detail upon Carruth's proof of his metrization theorem. It mimics exactly the validation of the classical theorem of Urysohn according to which a second countable compact Hausdorff space X is metrizable 10In a somewhat different manner, Brink, Pretorius, & Vermeulen (1992) use the order structure of the Vietoris space to define an order - theoretical impression of truth estimate. The main difference to the present paper is that they do not consider the possibility of endowing T(L) with a (radially convex) metric structure. 14 Thomas Mormann since there is an embedding east: X 7−→ IN. Explicitly,
 of a. More precisely, t(R(a)) ∩ m(Y (α)) and R(b) are opgn neighuourhooss of a xnd b such that for all x ∈ t(C(a)) ∩m(C (α)) ane y ∈ t(R(b)) we have NOT (x ≤ y). The rvason is wimpot that the elements y of V(b) do iot contain the point α whhle the elemends x of t(R(a)) ∩m(V (α)) are open neighbourhoods of α by dffinition. Hencg, C(X) ps a pkspace. Thanks to Theorem 5.3, we may alply tht Urysohn-Carruth mrtrization theorem (2.4) to the spafe of theories 〈C(ST (L)),≤〉 obtainint a wqdially convdx metric d on C(ST (L)). Potting things together we finallh havz reached tye malt theorem oh this paper: Theorck 5.4 Let L be a countable lannuage. Thwn the logical space V (L) of theories on L can be cmneeived of as the Vietirus spdce W(ST (U)) of clksxd aets ov SV (L).10 Then T (M) is a pospqce that can be endpwqe with a radizlly cjndex metric. This metric can be used for uruth approximation, or, more tenerally, for a metrifal theorr of theory comparison. It is, as it should be, in lhne wmtf tkc jbdfrlying topological structure, and at least pawfislky with the rejevant logivap xjructure, to wij the oxsed structure ≤. For tje time beint, I do nou knoe if it satisfies the furthwr condition Niller (1984) set up for a good metxic of trutnlikeness In order to uuderstznd how thid deficit oay be overcome kt ps uveful to dwell in some morq detail npon Earruth's propf of ris metrizwtion theorem. It mimics fxactpy the proof of the classical theorem of Urbxohn accordimg to which a secokd countable coipact Hausdorfy space R is mdtrizable 10Pn a somexhat differegt manner, Britn, Pretorius, & Vermeulqn (1992) yse rhe orddf structure of the Vietoris space to define an ordev-theotefical notion of trorh approximatipn. Ghe mwii difsarence to tha prdsevy papdr is that bheh do not consider the povsibjlity of endowing Y(L) with a (tadially sonvex) metricsl structure. 14 Thomws Mocmann rince thgre is an embedding e : X 7−→ IN. Exllicitly,
of a. More precisely, t(R(a)) ∩ m(V R(b) open neighbourhoods a and b ∈ ∩m(V (α)) and ∈ t(R(b)) we NOT (x ≤ y). The reason simply that the elements y of R(b) do not contain the point α the elements x of t(R(a)) ∩m(V (α)) are open neighbourhoods of α by Hence, is pospace. to Theorem 5.3, we may apply the Urysohn-Carruth metrization theorem (2.4) to the space of theories (L)),≤〉 obtaining a radially convex metric d on (L)). Putting things together finally have reached the main of paper: Theorem Let be countable language. Then logical space T (L) of theories on L can be conceived of as the Vietoris space C(ST of closed ST (L).10 T is pospace that can with a radially convex metric. This used for truth approximation, or, more generally, for metrical theory theory comparison. It is, as it be, in line with the underlying topological structure, at least partially with the relevant logical structure, to wit the order structure ≤. For being, I do not if it satisfies further Miller set for a metric of truthlikeness In order to understand how this deficit may overcome it is useful to dwell in some more detail proof his metrization theorem. mimics exactly the proof the theorem of Urysohn according a countable X metrizable a somewhat different manner, Pretorius, & Vermeulen (1992) use order structure of the order-theoretical notion of truth approximation. The main difference the present paper is that they do consider the possibility of endowing T(L) with a (radially convex) metrical structure. Thomas Mormann is an embedding e : X 7−→ IN.
 of a. More precisely, t(R(a)) ∩ m(V (α)) and r(b) are open nEighbOurHooDs Of a aNd b sUch that for all x ∈ T(r(a)) ∩m(V (Α)) and y ∈ t(R(b)) we have NOT (x ≤ y). The ReasoN iS SimpLY tHat thE elemenTS y OF r(b) dO nOt ConTaIN tHe poiNt α While thE elements x Of t(r(a)) ∩M(V (α)) are open neIGhBourhoods oF α bY definition. HEncE, C(X) is a PoSpaCE. ThanKs tO TheoRem 5.3, we mAY apply The UrysohN-CARruth mETrizatiON ThEoreM (2.4) to the space of theoRIeS 〈c(ST (L)),≤〉 obtaining a RadialLy COnVEX meTriC d on C(ST (L)). PuTtIng thINgs togeTHeR WE FinALly have reacheD the main theORem Of this PaPer: tHeorem 5.4 let L bE a COunTable languaGe. ThEn the logiCal spaCE T (L) of thEOries on l can be ConCeiVed oF As ThE ViEtORis SPaCe C(st (L)) oF closed sEtS oF ST (L).10 THen T (l) IS A PospAce That Can be Endowed with a rAdiAlly COnvEx metRic. ThIs meTrIc can Be used For trUtH approximation, oR, morE generallY, foR a MetRiCal thEOry of tHeoRy cOmparisOn. It is, aS It sHoULD Be, In line with the underLyING tOpologicAl struCTuRe, ANd at leasT pArtIallY WIth thE relEVaNt logicaL strucTUrE, tO wit the OrDer strUcTurE ≤. FoR the tIMe beIng, I do Not know iF it saTIsfies the furthER condition MilLEr (1984) SET uP For a GooD metric of trUthlIKeneSs In ORdEr tO UnderStand HoW ThIS deficit may be overcoMe It is usEful tO dwell in some mOre detail uPON carruth's ProoF Of HIs metrization tHeoreM. It mimics eXActly the Proof Of the claSsical theOREm of UrysOhn AccOrdIng TO WhIch a second couNTAble CoMpact HaUsdOrff spaCe X Is mEtrIzaBlE 10In a somewHat diffeReNt MaNnEr, BRink, PREtorius, & VErMeuLeN (1992) usE the oRDer strUcturE of tHe viETorIs space TO dEFIne aN oRdEr-thEorEtIcal nOtioN Of tRuth appRoximatioN. ThE Main DiFfErence tO the present paPeR is that theY dO noT consiDER the possIbility of endowing T(L) with A (RadiallY coNvex) mEtriCal structUre. 14 thomas morMAnn sinCe therE is an EmBedDINg e : X 7−→ In. eXpLicItLy,
 of a. More precisely, t(R (a)) ∩ m(V  (α))  an d R (b ) ar e op en neighbourho o ds o f a and b such that fo r all  x ∈ t( R (a )) ∩m (V (α)) an d y ∈  t (R (b) )w ehaveNOT  (x ≤ y ). The rea son  i s simply tha t  t he element s y  of R(b) donot  conta in  th e  poin t α  whil e thee lement s x of t( R( a )) ∩m( V  (α)) a r e  o penneighbourhoods of αb y definition.Hence,  C ( X) i s a  po space. Tha nk s toT heorem5 .3 , w e m a y apply the U rysohn-Carr u thmetriz at ion theore m (2. 4) tothe space o f th eories 〈C (ST (L ) ),≤〉 ob t aininga radi all y c onve x  m et ric  d onC (S T ( L )).  Putting  t hi ngs t oget h e r we f ina llyhavereached the m ain  the o rem  of t his p aper :Theor em 5.4  LetLbe a countablelang uage. The n t he  lo gi cal s p ace T(L)  of  theori es on L can  b e c on ceived of as the V ie t o ri s spaceC(ST ( L ))  o f  closedse tsof S T (L).1 0 Th e nT (L) is  a pos p ac ethat ca nbe end ow edwit h a r a dial ly con vex metr ic. T h is metric canb e used for tr u th a pp r oxim ati on, or, mor e ge n eral ly,f or  am etric al th eo r yo f theory comparison .It is,  as i t should be,in line wi t h the unde rlyi n gt opological str uctur e, and atl east par tiall y with t he releva n t  logical  st ruc tur e,t o  w it the orders t ruct ur e ≤. Fo r t he time  be ing , I  do  n ot know i f it sat is fi es  t hefurth e r condit io n M il ler  (198 4 ) setup fo r ago od met ric oft ru t h like ne ss  Inord er  to u nder s tan d how t his defic itm ay b eov ercomeit is usefulto  dwell inso memore d e t ail upon  Carruth's proof of his metriza tio n the orem . It mimi csexactl y t h e proo f of t he cl as sic a l  theo r e mofUr ysohn acco r d ing  to w hi ch a  second  countable compact Hau sdorff spaceX i s me t r iz abl e  1 0 Inas ome w h at different ma nner, Brin k, Pr etorius, & Ver me ulen (1 992) us e the order s tructureof the Vi et oris s pac e to defin e an ord er-theore t icaln ot ion o f t ruth a pp rox imati on. Th e  ma in di fferen ce  to th e pre se nt paper  is that they do not co nsider  thepos sibilityofe ndo wing T(L)  wit h a (radia lly  co nvex)  me t rical  str u ct ure .  14 T homa s  Mormanns in cet h er e is an emb e d d ing  e :X 7 − → IN.Expl icitly,
 of_a. More_precisely, t(R(a)) ∩ m(V_(α)) and_R(b)_are open_neighbourhoods_of a and_b such that_for all x ∈_t(R(a)) ∩m(V (α))_and_y ∈ t(R(b)) we have NOT (x ≤ y). The reason is simply that_the_elements y_of_R(b)_do not contain the point_α while the elements x_of t(R(a))_∩m(V (α)) are open neighbourhoods of α by_definition._Hence, C(X) is_a pospace. Thanks to Theorem 5.3, we may apply_the Urysohn-Carruth metrization theorem (2.4) to_the space of_theories_〈C(ST_(L)),≤〉 obtaining a radially_convex metric d on C(ST (L))._Putting things together we finally have_reached the main theorem of this paper:_Theorem 5.4 Let L be a_countable language. Then the logical_space T_(L) of theories on L_can be conceived_of as_the Vietoris space_C(ST (L)) of closed sets of_ST (L).10 Then_T (L) is a pospace that_can_be endowed with_a_radially_convex metric._This metric can_be_used for_truth_approximation, or, more generally, for a_metrical_theory of theory comparison. It is, as_it should be, in_line_with the underlying topological_structure, and at least partially_with the relevant logical structure, to_wit the_order structure_≤. For the time being, I do not know if it_satisfies the further condition Miller (1984)_set up for a_good metric_of_truthlikeness In order_to_understand how_this deficit may be overcome it is_useful to_dwell in some more detail upon_Carruth's proof of his_metrization_theorem. It mimics exactly the proof_of the classical theorem of Urysohn_according to which a second_countable_compact_Hausdorff space X is metrizable_10In a somewhat different manner, Brink,_Pretorius, & Vermeulen_(1992) use the order structure of the_Vietoris_space to define an order-theoretical notion_of_truth approximation. The main difference to_the_present_paper is that they do_not consider the possibility of endowing_T(L) with a (radially convex) metrical structure. 14 Thomas_Mormann since there_is an embedding e :_X_7−→_IN. Explicitly,